{"title":"权值为整数的无向图中的所有对最短路径","authors":"Avi Shoshan, Uri Zwick","doi":"10.1109/SFFCS.1999.814635","DOIUrl":null,"url":null,"abstract":"We show that the all pairs shortest paths (APSP) problem for undirected graphs with integer edge weights taken from the range {1, 2, ..., M} can be solved using only a logarithmic number of distance products of matrices with elements in the range (1, 2, ..., M). As a result, we get an algorithm for the APSP problem in such graphs that runs in O~(Mn/sup /spl omega//) time, where n is the number of vertices in the input graph, M is the largest edge weight in the graph, and /spl omega/<2.376 is the exponent of matrix multiplication. This improves, and also simplifies, an O~(M/sup (/spl omega/+1)/2/n/sup /spl omega//) time algorithm of Galil and Margalit (1997).","PeriodicalId":385047,"journal":{"name":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1999-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"142","resultStr":"{\"title\":\"All pairs shortest paths in undirected graphs with integer weights\",\"authors\":\"Avi Shoshan, Uri Zwick\",\"doi\":\"10.1109/SFFCS.1999.814635\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that the all pairs shortest paths (APSP) problem for undirected graphs with integer edge weights taken from the range {1, 2, ..., M} can be solved using only a logarithmic number of distance products of matrices with elements in the range (1, 2, ..., M). As a result, we get an algorithm for the APSP problem in such graphs that runs in O~(Mn/sup /spl omega//) time, where n is the number of vertices in the input graph, M is the largest edge weight in the graph, and /spl omega/<2.376 is the exponent of matrix multiplication. This improves, and also simplifies, an O~(M/sup (/spl omega/+1)/2/n/sup /spl omega//) time algorithm of Galil and Margalit (1997).\",\"PeriodicalId\":385047,\"journal\":{\"name\":\"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)\",\"volume\":\"6 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1999-10-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"142\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/SFFCS.1999.814635\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/SFFCS.1999.814635","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
All pairs shortest paths in undirected graphs with integer weights
We show that the all pairs shortest paths (APSP) problem for undirected graphs with integer edge weights taken from the range {1, 2, ..., M} can be solved using only a logarithmic number of distance products of matrices with elements in the range (1, 2, ..., M). As a result, we get an algorithm for the APSP problem in such graphs that runs in O~(Mn/sup /spl omega//) time, where n is the number of vertices in the input graph, M is the largest edge weight in the graph, and /spl omega/<2.376 is the exponent of matrix multiplication. This improves, and also simplifies, an O~(M/sup (/spl omega/+1)/2/n/sup /spl omega//) time algorithm of Galil and Margalit (1997).
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