Yijie Han

Shortest paths on a polyhedron

We present an algorithm for determining the shortest path between a source point and any destination point along the surface of a polyhedron (need not be convex). Our algorithm uses a new approach which deviates from the conventional “continuous Dijkstra” technique. It takes <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt>) time and ⊖(<italic>n</italic>) space to determine the shortest path and to compute the inward layout which can be used to construct a structure for processing queries of shortest path from the source point to any destination point.

SHORTEST PATHS ON A POLYHEDRON, Part I: COMPUTING SHORTEST PATHS

We present an algorithm for determining the shortest path between any two points along the surface of a polyhedron which need not be convex. This algorithm also computes for any source point on the surface of a polyhedron the inward layout and the subdivision of the polyhedron which can be used for processing queries of shortest paths between the source point and any destination point. Our algorithm uses a new approach which deviates from the conventional “continuous Dijkstra” technique. Our algorithm has time complexity O(n2) and space complexity Θ(n).

Concurrent threads and optimal parallel minimum spanning trees algorithm

This paper resolves a long-standing open problem on whether the concurrent write capability of parallel random access machine (PRAM) is essential for solving fundamental graph problems like connected components and minimum spanning trees in <italic>O</italic>(log<italic>n</italic>) time. Specifically, we present a new algorithm to solve these problems in <italic>O</italic>(log<italic>n</italic>) time using a linear number of processors on the exclusive-read exclusive-write PRAM. The logarithmic time bound is actually optimal since it is well known that even computing the “OR” of <italic>n</italic>bit requires &OHgr;(log <italic>n</italic> time on the exclusive-write PRAM. The efficiency achieved by the new algorithm is based on a new schedule which can exploit a high degree of parallelism.

Improved algorithm for all pairs shortest paths

We present an improved algorithm for all pairs shortest paths. For a graph of n vertices our algorithm runs in O(n3(log log n/log n)5/7) time. This improves the best previous result which runs in O(n3(log log n/log n)1/2) time.

An O ( n 3 (log log n /log n ) 5/4 ) Time Algorithm for All Pairs Shortest Path

AbstractnWe present an O(n3(logu2009logu2009n/logu2009n)5/4) time algorithm for all pairs shortest paths. This algorithm improves on the best previous result of O(n3/logu2009n) time.n

Improved Fast Integer Sorting in Linear Space

We present improved fast deterministic algorithm for integer sorting in linear space. Our algorithm sorts <i>n</i> integers in linear space in <i>&Ogr;</i>(<i>n</i> log log <i>n</i> log log log <i>n</i>) time. This improves the <i>&Ogr;</i>(<i>n</i>(log log <i>n</i>)^3/2) time bound given in [6]. When the <i>n</i> integers in {0,1,…, <i>m</i> - 1} to be sorted satisfying log <i>m</i> ⪈(log <i>n</i>)²+∈, 0 < ∈ < 1, the time complexity for sorting can be further reduced to <i>&Ogr;</i>(<i>n</i> log log <i>n</i>). These results are obtained by applying signature sorting on our previous result[6].

Efficient parallel algorithms for computing all pair shortest paths in directed graphs

We present parallel algorithms for computing all pair shortest paths in directed graphs. Our algorithm has time complexityO(f(n)/p+I(n)logn) on the PRAM usingp processors, whereI(n) is logn on the EREW PRAM, log logn on the CCRW PRAM,f(n) iso(n3). On the randomized CRCW PRAM we are able to achieve time complexityO(n3/p+logn) usingp processors.

Mapping a chain task to chained processors

Abstract We present algorithms for computing an optimal mapping of a chain task to chained processors. Our algorithm has time complexity no larger than O( m + p 1+ e ) for any small e>0. This represents an improvement over the recent best results of an O( mp ) algorithm and an O( m + p 2 log 2 m ) algorithm. We extend our algorithm to the case of mapping a ring task to a ring of processors with time complexity O( m 1+ e ) and the case of mapping multiple chain tasks to chained processors with time complexity O( mn + p 1+ e ).

Matching partition a linked list and its optimization

We show the curve O( n log i p + log n + log i) for the time complexity of computing a maximal matching for a linked list, where n is the size of the input list, p is the number of processors used in the algorithm and i is an adjustable parameter. For all constructible i the time complexity represented by the curve can be realized. Our algorithm is optimal using up to O( n log n ) processors with an arbitrarily large constant i. This algorithm can be used to compute a maximal independent set or a 3 coloring for a linked list.

Parallel Integer Sorting Is More Efficient Than Parallel Comparison Sorting on Exclusive Write PRAMs

We present a significant improvement for parallel integer sorting. On the EREW (exclusive read exclusive write) PRAM our algorithm sorts n integers in the range {0,1, . . . ,m-1 } in time O(log n) with