CS253: Shortest Path (2019)

Shortest Path Algorithm Data Structures and Algorithms Emory University Jinho
D. Choi

Emory University Logo Guidelines #PKPUVKVWVKQPCUNCTIGCPFXCTKGFCU'OQT[TGSWKTGUCEQPUKUVGPVXKUWCNKFGPVKV[VJCVWPKƂGUKVUXCTKQWU CHƂNKCVGU'OQT[oUEWTTGPVUVCPFCTFUYJKEJJCXGDGGPKPWUGUKPEGTGKPHQTEGVJGWPKSWGEJCTCEVGT CPFSWCNKV[QHGCEJCECFGOKECPFCFOKPKUVTCVKXGWPKVYJKNGUKOWNVCPGQWUN[OCMKPIKVENGCTVJCV'OQT[ UVCPFUDGJKPFGCEJQHVJGO +PCFFKVKQPVQVJGOCKP7PKXGTUKV[ITCRJKEKFGPVKƂGTUOQUVUEJQQNUCPFOCLQTWPKVUJCXGVJGKTQYP EQORNGOGPVCT[UGVQHKFGPVKV[ITCRJKEUHQTRTKPVCPFYGDYJKEJYGTGFGXGNQRGFKPECTGHWNEQPUWNVC-
VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 S T 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous S T 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T Visited 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 (5, 0) ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 5 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 (3, 3) 5 5 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 (3, 3) 5 5 9 5 (4, 9) 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 5 9 5 (4, 9) 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 13 5 3 (1, 13) 9 5 (4, 9) 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 13 5 3 (1, 13) (4, 7) 9 7 5 3 (4, 9) 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 13 5 3 (1, 13) 4 9 7 5 3 (4, 9) 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 13 5 3 (1, 13) 4 9 7 5 3 (4, 9) 10 4 (2, 10) 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 13 5 3 (1, 13) 4 9 7 5 3 10 4 (2, 10) 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 13 5 3 (1, 13) 4 9 7 5 3 10 4 2 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 13 5 3 (1, 13) 4 9 7 5 3 10 4 2 12 2 (0, 12) 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 13 5 3 (1, 13) 4 9 7 5 3 10 4 2 12 2 0 2 3 3 4 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 2 0 1 2 3 4 5 4 5 10 9 ∞ ∞ ∞ ∞ ∞ ∞ 0 1 2 3 4 5 Distance ∅ ∅ ∅ ∅ ∅ ∅ Previous Queue S T 0 Visited 3 5 3 13 5 3 4 9 7 5 3 10 4 2 12 2 0 2 3 3 4 1 Dijkstra's Algorithm

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited 2 1 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 2 1 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 (3, 0) 2 1 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 2 1 3 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 2 1 3 1 3 (1, 1) 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 2 1 3 1 3 (1, 1) 2 3 (2, 2) 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 2 1 3 1 3 2 3 (2, 2) 1 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 2 1 3 1 3 2 3 (2, 2) 1 3 1 (0, 3) 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 2 1 3 1 3 2 3 1 3 1 (0, 3) 2 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 3 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 2 1 3 1 3 2 3 1 3 1 2 0 1 2 4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited 1 1 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 1 1 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 (3, 0) 1 1 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 1 1 3 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 1 1 3 1 3 (1, 1) 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 1 1 3 1 3 (1, 1) 2 3 (2, 2) 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 1 1 3 1 3 2 3 (2, 2) 1 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 1 1 3 1 3 2 3 (2, 2) 1 3 1 (0, 3) 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 1 1 3 1 3 2 3 1 3 1 (0, 3) 2 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 1 1 3 1 3 2 3 1 3 1 (0, 3) 2 -2 2 (1, -2) 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 1 1 3 1 3 2 3 1 3 1 (0, 3) 2 -2 2 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 4 1 0 3 2 ∅ ∅ ∅ ∅ Previous ∞ ∞ ∞ ∞ Distance 0 1 2 3 Queue Visited T S 0 1 1 3 1 3 2 3 1 3 1 2 0 -2 2 2 2 -4

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Vertex Distance Pair 5 private class VertexDistancePair implements Comparable<VertexDistancePair> { public int vertex; public double distance; public VertexDistancePair(int vertex, double distance) { this.vertex = vertex; this.distance = distance; } @Override public int compareTo(VertexDistancePair p) { double diff = this.distance - p.distance; if (diff > 0) return 1; if (diff < 0) return -1; return 0; } }

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 6 public Integer[] getShortestPath(Graph graph, int source, int target) { PriorityQueue<VertexDistancePair> queue = new PriorityQueue<>(); Integer[] previous = new Integer[graph.size()]; double[] distances = new double[graph.size()]; Set<Integer> visited = new HashSet<>(); init(distances, previous, target); queue.add(new VertexDistancePair(target, distances[target])); while (!queue.isEmpty()) { // To be filled. } return previous; }

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's Algorithm 7 private void init(double[] distances, Integer[] previous, int target) { for (int i=0; i<distances.length; i++) { if (i == target) distances[i] = 0; else { distances[i] = Double.MAX_VALUE; previous[i] = null; } } }

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 8 while (!queue.isEmpty()) { VertexDistancePair u = queue.poll(); visited.add(u.vertex); for (Edge edge : graph.getIncomingEdges(u.vertex)) { int v = edge.getSource(); if (!visited.contains(v)) { double dist = distances[u.vertex] + edge.getWeight(); if (dist < distances[v]) { distances[v] = dist; previous [v] = u.vertex; queue.add(new VertexDistancePair(v, dist)); } } } }

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 8 while (!queue.isEmpty()) { VertexDistancePair u = queue.poll(); visited.add(u.vertex); for (Edge edge : graph.getIncomingEdges(u.vertex)) { int v = edge.getSource(); if (!visited.contains(v)) { double dist = distances[u.vertex] + edge.getWeight(); if (dist < distances[v]) { distances[v] = dist; previous [v] = u.vertex; queue.add(new VertexDistancePair(v, dist)); } } } } Complexity?

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's vs A* 9 private void init(double[] distances, Integer[] previous, int target) { for (int i=0; i<distances.length; i++) { if (i == target) distances[i] = 0; else { distances[i] = Double.MAX_VALUE; previous[i] = null; } } }

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra's vs A* 9 private void init(double[] distances, Integer[] previous, int target) { for (int i=0; i<distances.length; i++) { if (i == target) distances[i] = 0; else { distances[i] = Double.MAX_VALUE; previous[i] = null; } } } + heuristic(i, target);

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 10 while (!queue.isEmpty()) { VertexDistancePair u = queue.poll(); visited.add(u.vertex); for (Edge edge : graph.getIncomingEdges(u.vertex)) { int v = edge.getSource(); if (!visited.contains(v)) { double dist = distances[u.vertex] + edge.getWeight(); if (dist < distances[v]) { distances[v] = dist; previous [v] = u.vertex; queue.add(new VertexDistancePair(v, dist)); } } } }

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW 10 while (!queue.isEmpty()) { VertexDistancePair u = queue.poll(); visited.add(u.vertex); for (Edge edge : graph.getIncomingEdges(u.vertex)) { int v = edge.getSource(); if (!visited.contains(v)) { double dist = distances[u.vertex] + edge.getWeight(); if (dist < distances[v]) { distances[v] = dist; previous [v] = u.vertex; queue.add(new VertexDistancePair(v, dist)); } } } } + heuristic(v, target);

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW A* Abstract Class 11 public abstract class AStar { public Integer[] getShortestPath(Graph graph, int source, int target) { PriorityQueue<VertexDistancePair> queue = new PriorityQueue<>(); Integer[] previous = new Integer[graph.size()]; double[] distances = new double[graph.size()]; Set<Integer> visited = new HashSet<>(); ... return previous; } } protected abstract double heuristic(int source, int target);

VKQPYKVJFGCPUCPFWPKVJGCFU&QYPNQCFCDNGNQIQUCPFYQTFOCTMUHQT'OQT[7PKXGTUKV[VJGUEJQQNU CPFOCLQTWPKVUECPDGHQWPFQPVJGYGDCVJVVRKFGPVKV[GOQT[GFW RTKPVITCRJKEUVCPFCTFU CPF JVVRYGDIWKFGGOQT[GFW YGDITCRJKEUVCPFCTFU 6JG'/14;YQTFOCTMKUCHGFGTCNN[TGIKUVGTGF VTCFGOCTM#UCPEVKQPGFKFGPVKƂGTQHVJG7PKXGTUKV[tCUEJQQNQTOCLQTWPKVNQIQVJCVKPENWFGUVJG UJKGNFU[ODQNCPFVJGYQTFOCTM'/14;tUJQWNFCRRGCTQPGCEJRWDNKECVKQP+H[QWYKUJVQJCXGCP KFGPVKƂGT WPKVUKIPCVWTG ETGCVGFURGEKƂECNN[HQT[QWTRTQITCOQTFGRCTVOGPVRNGCUGEQPVCEVVJG1HƂEG QH$TCPF/CPCIGOGPVCVQTUVCPKUMQFOCP"GOQT[GFW 6JGV[RGHCEG)QWF[KUTGUGTXGFHQTVJG'OQT[VTCFGOCTMUCPFPGXGTUJQWNFDGWUGFKPVGZVQTFKURNC[ EQR[;QWPGXGTUJQWNFCVVGORVVQTGPFGTVJG'OQT[NQIQD[V[RKPIVJGNGVVGTUKPCYQTFRTQEGUUKPIQT RCIGNC[QWVRTQITCO0QPGQHVJGNQIQUKUCV[RGFYQTFDWVTCVJGTKUURGEKƂECNN[FGUKIPGFXGEVQTCTV 'OQT[VKGTQPGNQIQUUJQWNFTGRTQFWEGQPN[KP'OQT[DNWG 2/5 DNCEMQTYJKVG6JGUJKGNFJGKIJV UJQWNFTGRTQFWEGCVqQTNCTIGT#IGPGTCNTWNGHQTURCEKPICTQWPFCP'OQT[NQIQKUVQKPVGITCVGCP QDXKQWUXKUWCNUGRCTCVKQPtPQFGUKIPGNGOGPVQTVGZVUJQWNFDGPGUVGFYKVJ'OQT[NQIQU 'OQT[oURTKOCT[EQNQTUCTG'OQT[DNWG 2/5 CPF[GNNQY 2/5 'OQT[7PKXGTUKV[YQTFOCTMU ECPDGTGRTQFWEGFKP'OQT[DNWG 2/5 DNCEMQTYJKVGQPCP'OQT[DNWGQTFCTMDCEMITQWPF YYYKFGPVKV[GOQT[GFW JVVRYGDIWKFGGOQT[GFW Dijkstra’s Algorithm 12 public class Dijkstra extends AStar { @Override protected double heuristic(int source, int target) { return 0; } }

CS253: Shortest Path (2019)

CS253: Shortest Path (2019)

Jinho D. Choi

More Decks by Jinho D. Choi

Featured

Transcript

Shortest Path Algorithm Data Structures and Algorithms Emory University Jinho