Improved Randomized Algorithms for Path Problems in Graphs PhD Thesis Surender Baswana Department of Computer Science & Engineering, I.I.T. Delhi Research supported in parts by a PhD fellowship from Infosys Technologies Ltd. Bangalore

Organization of the talk  Practical importance of algorithms/theoretical computer science.  Description of the problems addressed during PhD.  Application of the problems, and new results.  What I gained from my PhD.  My experiences, esp. invaluable support from Infosys.

Theoretical Computer Science  Is it feasible to solve a problem ?  Is a problem practically tractable ? Traveling Salesman Problem : Given a set of cities, find the tour of least distance. No algorithm that performs better than 2 n operations.  Designing algorithm for solving a problem that requires minimum time (# of operations) and optimal storage.

Sorting Algorithms  Selection sort, Bubble sort, Insertion sort n 2 comparisons in worst case  Merge Sort : n log 2 n comparisons in worst case  Quick sort : (A randomized algorithm) n 2 comparisons in worst case n log 2 n comparisons on an average  Quick sort performs better than all the sorting algorithms Practically.  Probability that quick sort deviates from its expected running time asymptotically is less than 2 –cn.

Path Problems in Graphs Process a given graph G(V,E) so that we can retrieve a path with certain characteristics between each pair of vertices. n = number of vertices in a given graph. m = number of edges in a given graph.

A Directed Graph

A Directed Weighted Graph

All Pairs Reachability (Transitive Closure) Given a pair of vertices x,y in V, Is there a path from x to y ?

All-Pairs Reachability (Transitive Closure) x y

x y Processing time : mn

All-Pairs Shortest Paths Given a pair of vertices x,y in V, report the shortest path (or distance) from x to y.

All-Pairs Shortest Paths x y a b c d e

All-Pairs Shortest Paths x y a b c d e x-b-e-y x-b-c-d-e-y x-a-y Processing time : mn log n

A Network Design Problem

Objective To minimize the number of links to be selected without increasing the route time between any pair of nodes too much.

An Example

An Example X Y A B C F

k-Spanner of a Weighted Graph Given a graph G(V,E), a sub-graph G S (V,E’) is said to be k-spanner if for each u,v in V, d s (u,v) ≤ k. d(u,v)

Application of Sparse Spanner  Network design, especially for making succinct routing tables.  Computational biology : in the reconstruction of phylogeny trees from matrices whose entries represent distances among contemporary living species.  Design of network synchronizers, and computation of all-pairs approximate shortest paths.

Computing a Sparse k-Spanner  Objective : a Sparse spanner with minimum stretch  A trade off between stretch and the size  A k-spanner in the worst case may need at-least n 1+1/k edges.  Example : There can’t be a 3-spanner with less than n 3/2 edges for every graph.

Algorithms for Computing (2k-1)-Spanner PersonsReferenceStretchRunning time Das, et al.Journal : DCG, 93 (2k-1) mn 1+1/k Thorup and Zwick Conf. : ACM STOC, 2001 (2k-1) mn 1/k

Algorithms for computing (2k-1)-spanner PersonsReferenceStretchRunning time Das, et al.Journal : DCG, 93 (2k-1) mn 1+1/k Thorup and Zwick Conf. : ACM STOC, 2001 (2k-1) mn 1/k Baswana and Sen Conf. : ICALP, 2003 (2k-1) m

Approximate Distance Oracles  Usually all-pairs shortest distances are stored in an nхn matrix.  Is it possible to require less than n 2 storage and still answer a distance query in constant time?  Yes, but at the expense of introducing approximation.  3-Approximate Distance oracle of size n 3/2

Advantage of Approximate Distance Oracles  For answering shortest path queries in external-memory

Advantage of Approximate Distance Oracles  For answering shortest path queries in external-memory RAM Hard-DiskCPU FastSlow

Approximate Distance Oracles  Thorup and Zwick [STOC’01] gave first algorithm for computing approximate distance oracle with Size : n 1+1/k Stretch : 2k-1  Preporcessing time for building a (2k-1)-approximate distance oracle : mn 1/k  Worst case preprocessing time for 3-Approximate Distance Oracle would be n 2.5.  Question : Can 3-approximate distance oracle be computed in n 2 time ?

Algorithms for Computing All-Pairs Approximate Distance Oracle PersonReference Stretch Size Preprocessing time DijkstraAll text Books on Algorithms 1n2n2 mn

Algorithms for Computing All-Pairs Approximate Distance Oracle PersonReference Stretch Size Preprocessing time DijkstraAll text Books on Algorithms 1n2n2 mn Thorup and Zwick Conf. ACM-STOC k-1n 1+1/k mn 1/k

Algorithms for Computing All-Pairs Approximate Distance Oracle PersonReference Stretch Size Preprocessing time DijkstraAll text Books on Algorithms 1n2n2 mn Thorup and Zwick Conf. ACM-STOC k-1n 1+1/k mn 1/k Baswana and Sen Conf. ACM-SIAM SODA k-1n 1+1/k Min (mn 1/k,n 2 )

Path Problems in Dynamic Graphs

Example of a Dynamic Graph x y b c d e f g

x y b c d e f g

x y b c d e f g

x y b c d e f g

x y b c d e f g

A Dynamic Graph Problem  Given a graph G=(V,E)  Online sequence of queries interspersed with updates.  Objective : 1. To Answer each query efficiently in online fashion. 2. Perform updates in the solution efficiently.  Naive Solution : After every update, re-compute the solution right from scratch.

A Dynamic Graph Algorithm  Maintain some data-structure that can answer each query efficiently.  The time required to maintain the data-structure after each update is much less than the static graph algorithm.  For each static graph algorithm/problem, there is a dynamic graph algorithm/problem as its counterpart.

Applications of Dynamic Graph Algorithms  Communication networks.  Relational databases augmented with transitive closure.  Incremental parser generator.

Maintaining All-Pairs Reachability under Deletion of edges  Static algorithm requires mn time.  A combination of algorithms of Demetrescu et al. [IEEE FOCS 2000] and Leeuwen et al. [TCS 1991] requires n 3/2 update time per edge deletion  Baswana S., Hariharan R., Sen S. design an algorithm that requires n 4/3 update time per edge deletion

Maintaining All-Pairs Shortest Paths under Deletion of Edges  Static algorithm requires mn time.  An algorithm of Demetrescu and Italiano [IEEE FOCS 2001] requires n 3 /m update time per edge deletion  For sparse graphs, the update time of above algorithm : n 2  Baswana S., Hariharan R., Sen S. design an algorithm that requires n 3/2 update time per edge deletion for all graphs  Proceedings of 34 th Annual ACM Symposium on Theory of Computing, Montreal, Quebec, Canada, May 19-21, 2002.

Maintaining All-Pairs Approximate Shortest Paths under Deletion of Edges  Improved static algorithms for shortest paths at the expense of approximation.  Can we achieve better update time for maintaining approximate shortest paths?  Baswana S., Hariharan R., Sen S. design an algorithm that requires Close to n update time per edge deletion  Proceedings of 14 th Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, Maryland, USA, January 12-14, 2003.

Maintaining All-Pairs Approximate Shortest Paths under Deletion of Edges Stretch factorUpdate time 3n 10/9 5n 14/13 7n 28/27

A Linear time Algorithm for computing a 3-spanner  Objective : Select no more than n 3/2 edges and ensure a stretch of 3 at-most.  Vertices of degree more than n ½ are difficult to handle.  Solution : * Clustering of vertices * Random sampling to form clusters

A Linear time Algorithm for computing a 3-spanner.  Sample n ½ vertices uniformly randomly  If a vertex v is not adjacent to any sample vertex, select all its edges for spanner.  Otherwise, add all the edges having weight less than that of the edge between the vertex and its nearest sampled vertex n v.  The vertex v belongs to the cluster centered at n v

The Linear time Algorithm contd. v

v

Analyzing the correctness of algorithm  Consider an edge u-v not present in the spanner.  Both u and v must belong to clusters.  Case1 : both u and v belong to same cluster  Case2 : both u and v belong to different clusters u v w v u w u’

Gains from my PhD  Fulfilled the desire of working in theoretical computer science.  A beginning of research.  The joy of exploring a research area of my interest independently.  Analytical skills that may be applied in diverse areas.

Invaluable support from Infosys  Every research need fulfilled.  Used the generous contingency grant for buying a few of my dream books.  Used the support for traveling to conferences abroad.  PhD won’t have been completed so peacefully without Infosys Fellowship.