Another story on Multi-commodity Flows and its “dual” Network Monitoring Rohit Khandekar IBM Watson Joint work with Baruch Awerbuch JHU TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A AAAA A A A
Outline Crash course: –Set cover problem and the greedy algorithm –Framework for distributed covering problems The maximum multi-commodity problem and its dual passive commodity monitoring problem Fast converging distributed approximation schemes
The Set Cover Problem Given a set of elements U subsets S 1, S 2, …, S k µ U with costs c 1, c 2, …, c k ¸ 0 Find Minimum cost collection of subsets whose union is entire U.
The Greedy Algorithm Gives O(log n) approximation where n = |U|. (r e = 1 if e is not yet covered)
The Fractional Set Cover Problem The LP relaxation of the set cover IP.
The Fractional Greedy Algorithm Gives O(log n) ( 1 + ² ) approximation. Drawback: #iterations = n/ ² 2
The Fractional Greedy Algorithm all
The Fractional Distributed Algorithm # iterations = Luby-Nissan (93), Garg-Konemann (98), Young (01) Also computes a near-optimum dual solution
Concurrent Multi-commodity Flow c e = capacity Maximum Throughput
Concurrent Multi-commodity Flow Send maximum total flow between the pairs subject to the edge-capacity constraints. Maximum Throughput
Concurrent Multi-commodity Flow Send maximum total flow between the pairs subject to the edge-capacity constraints. Maximum Throughput
Distributed Computation Model The ROUTERS model: “Intelligence” is embodied in the network routers Computations takes place by exchanging messages between neighboring routers Complexity measures: Approximation ratio ((1+ ² ) approximation) Message congestion (# messages/router/round) Space complexity (space needed/router) Convergence time (# rounds to converge) Computational complexity (total work)
Multicommodity Problem & Its Dual dual = set cover edges = sets paths = elements Dual: Probe edges e with frequency x e so that each path gets probed to an extent 1 while minimizing the total cost of probing e c e x e Passive commodity monitoring
Main Result There is an algorithm for maximum multicommodity flows and passive commodity monitoring with the following properties approximation convergence space and messages/router computational overhead L = maximum hop-length of a flowpath
Comparison with Previous Work
The Algorithm Set cover with edges as sets and paths as elements Associate with each path p, a residual requirement (profit of path p) ( ® is a constant)
The Algorithm Repeat: For all edges that (approximately) minimize the cost-to-profit ratio: increase Increase the flow on all paths through such edges
How to compute aaaaaaaa A shortest path algorithm (Dijkstra) computes: Compute A similar (dynamic programming) algorithm computes: Computing shortest paths on a “semi-ring”
How to compute aaaaaaaa l1l1 l2l2 l3l3 l4l4 11 22 33 44
Conclusions First multi-commodity algorithm –Via dual multi-cut problem –Breaks the (m) convergence barrier –Convergence polynomial in path-length L Question: Can we get O(L) convergence?
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