CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

1. CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

2. Lecture 26 • Topic: • Maximal Independent Set • Sources: • Luby • Schneider & Wattenhofer • Linial • MIT 6.885 Fall 2008 slides Discrete Algs for Mobile Wireless Sys

3. Overview • Recall that a minimum connected dominating set is a useful substructure of a graph representing a network: • routing • medium access control • coverage • Computing a MCDS in a general graph is NP-complete • What about special-case graphs that still reflect the reality of wireless networks? • UDG too restrictive • QUDG still too restrictive • let's try growth-bounded graphs (GBG), a.k.a. bounded independence graphs (BIG) Discrete Algs for Mobile Wireless Sys

4. Overview • In BIG model, a maximal independent set is a constant approximation of a MCDS • A MIS is an independent subset S of the nodes of a graph (none of the nodes in S are neighbors), and no superset of S is independent • [SW] paper gives an O(log*n) time algorithm for MIS in BIG model • log*n is number of times you can take the log of n until reaching 1 • algorithm is distributed, deterministic, and does not require location information • Running time is optimal (cf. paper by Linial) Discrete Algs for Mobile Wireless Sys

5. Unit Disk Graphs R R Wireless networks often modeled as unit disk graphs Discrete Algs for Mobile Wireless Sys

6. More Realistic Graphs Discrete Algs for Mobile Wireless Sys

7. Bounded Independence Graph Even more general than quasi unit disk graphs No links between far-away nodes Close nodes tend to be connected In particular: Densely covered area  many connections bounded neighborhood  bounded independent set Discrete Algs for Mobile Wireless Sys

8. Bounded Independence Graphs (BIGs) Definition: Given a function f(r), a graph G=(V,E) is f(r)-independence bounded if for all nodes v in V and all r ≥ 0, the size of a maximum IS in the r-neighborhood of v is at most f(r). Note that f is only a function of r and in particular independent of the number of nodes n. Discrete Algs for Mobile Wireless Sys

9. Bounded Independence Graphs (BIGs) Typically require that f(r) = poly(r). It can never be more than exponential. UDGs and QUDGs are independence-bounded with f(r) = O(r2). Discrete Algs for Mobile Wireless Sys

10. a maximal independent set a maximum independent set Maximal vs. Maximum IS Discrete Algs for Mobile Wireless Sys

11. MIS and DS • A MIS is a dominating set (DS) • If S is an IS but does not dominate some node, then the undominated node can be added to S while maintaining the independence property • But a DS is not necessarily independent • two dominators are allowed to be neighbors (not independent) Discrete Algs for Mobile Wireless Sys

12. MIS and MDS Theorem: On an f(r)-independence bounded graph G, a MIS is a f(1)-approximation of an MDS. Proof: Consider any maximal IS S of G. Suppose T is a minimum DS of G. • Every node in S is either in T or is a neighbor of some dominator t in T • Since G is a f(r)-BIG, t has at most f(1) elements of S as its neighbors • So |S| ≤ f(1) •|T| Discrete Algs for Mobile Wireless Sys

13. Distributed MIS Algorithm • For general graphs[Luby]: A simple parallel algorithm for the MIS problem(similar algorithm in [Alon,Babai,Itai]) • Randomized algorithm • Runs in O(log n) rounds in expectation and with high probability • Can we do better in special-case graphs? Discrete Algs for Mobile Wireless Sys

14. Log-Star MIS Algorithm for BIGs • Assumptions: • Every node has a unique ID between 1 and n • For simplicity, assume that all nodes know f(r) and n (not necessary) • For simplicity, synchronous model (not necessary) • Main result of [SW]:O(log*n) time MIS algorithm for bounded independence graph Discrete Algs for Mobile Wireless Sys

15. Algorithm: Basic Structure • During the algorithm, each node is always in one of 5 states: • competitor: Node actively competes to be in MIS • dominator: Node has joined the MIS • dominated: Node has a neighbor in the MIS, will definitely not join MIS • ruler: Node not actively in competition, will compete again actively if there are no neighboring competitors left • ruled: Neighbor of ruler, does not start competing again before all neighboring rulers become ruled themselves. Discrete Algs for Mobile Wireless Sys

16. Algorithm: Basic Structure • Algorithm consists of f(f(2) + 3) stages • Each stage consists of f(2) + 1 phases • Each phase consists of log*n + 2 competitions • Each competition needs a constant number of rounds • So total number of rounds is O(1)*(log*n + 2)*(f(2)+1)*f(f(2)+3) • which is O(log*n) since f(c) = O(1) when c = O(1) Discrete Algs for Mobile Wireless Sys

17. Competitions • Every competitor v starts a competition with a number rv and computes new rv’ • initially rv = ID(v) • Computation of rv’: • u: neighboring competitor with minimal ru • if ru> rvthen rv’ = 0 • else, rv’ is computed from the base-2 representations of rv and ru:rv’ is position of highest bit that is 1 in rv and 0 in ru(position of least significant bit is 1) Discrete Algs for Mobile Wireless Sys

18. Competitions • rv’ is position of highest bit that is 1 in rv and 0 in ru • position of least significant bit is 1 • Examples: • rv = (10100010)2, ru = (10010110)2 rv’ = 6 • rv = (00101000)2, ru = (00100101)2  rv’ = 4 Discrete Algs for Mobile Wireless Sys

19. Competition: New State • Compute new rv’ based on rv and min ru among neighboring competitors • Update state based on new values of v and neighbors: • If rv’ < ru’ for all neighboring competitors  v becomes dominator • Else if neighbor of v becomes dominator  v becomes dominated • Else if rv’ · ru’ for all neighboring competitors  v becomes ruler • Else if v has neighboring ruler  node becomes ruled • Else v stays competitor Discrete Algs for Mobile Wireless Sys

20. Competition: New State • Lemma: Dominators always form an independent set. • No 2 adjacent nodes can become dominator together. • Nodes that are dominated do not compete any further. • Only competing nodes can become dominator. Discrete Algs for Mobile Wireless Sys

21. Reducing the Competitors Lemma: After log*n + 2 competitions, no node is a competitor any more. Proof: • Initially, rv = ID(v), hence, rvuses at most log n bits • Hence, rv’ uses at most log log n bits • After log*n + 2 competitions, rvis in {0,1} • All nodes v with rv=0 become dominator or ruler • Neighbors become dominated or ruled • If rv=1 and all neighboring competitors u have ru=1, v becomes ruler Discrete Algs for Mobile Wireless Sys

22. Phase log*n+2 competitions are called a phase For next phase: • All rulers become competitors again • All rv are set back to ID(v) Discrete Algs for Mobile Wireless Sys

23. Stage Main technical lemma: No node becomes a ruler in the (f(2)+1)st phase. Thus, after f(2)+1 phases there are only nodes that are dominators, dominated, or ruled. Proof: Read the paper. f(2)+1 phases are called a stage. In new stage, ruled nodes become competitors again(note: there are no rulers any more…) Discrete Algs for Mobile Wireless Sys

24. Proof of Progress • Lemma: Let v be a competitor at the beginning of a stage. During the stage, a node at distance at most f(2)+1 becomes dominator. • Proof: • At the end of a stage, each node is ruled, dominated, or a dominator • Show that after i phases, there is a node at distance at most i that is not ruled Discrete Algs for Mobile Wireless Sys

25. Proof of Progress • Show that after i phases, there is a node at distance at most i that is not ruled • Induction on i: • Clear for i=0 (v is not ruled) • Let w be node that is not ruled at distance at most i after i phases • If w does not become ruled in (i+1)st phase, ok. • If w becomes ruled in a competition of the (i+1)st phase, some neighbor w’ becomes a ruler (w’ is at distance at most i+1). • w’ remains a ruler until the end of the phase and then becomes a competitor. Discrete Algs for Mobile Wireless Sys

26. Proof of Progress • After f(2) phases, there is a ruler at distance at most f(2) or a dominator at distance at most f(2)+1. • If it is a ruler, itself or a neighbor of it becomes dominator in phase f(2)+1. • Thus if v is a competitor at the beginning of a stage, then during the stage, a node at distance at most f(2)+1 becomes a dominator. Discrete Algs for Mobile Wireless Sys

27. Proof of Progress • Theorem: The algorithm terminates with a MIS after at most f(f(2)+1) stages. • Proof: • The algorithm terminates as soon as there are no ruled nodes at the end of a stage (i.e., all nodes are dominators or dominated) • Suppose in contradiction there is still a ruled node v after stage f(f(2)+1). • v was a competitor in all f(f(2)+1) stages. • In every stage, a node in (f(2)+1)-neighborhood of v joins the MIS • At most f(f(2)+1) nodes in (f(2)+1)-neighborhood of v can join an ind. set (because of BIG model) • Hence, the ind. set is maximal and v cannot be a competitor any more. Discrete Algs for Mobile Wireless Sys

28. Comments • In the paper, the algorithm is described in a way that does not require knowledge of f(r) and n • stages and phases need to be locally synchronized • algorithm works for all graphs, time complexity depends on graph • Algorithm is asymptotically optimal: • Result in [Linial]: Any deterministic algorithm needs at least (log*n) rounds to color a ring with O(1) colors. • From a c-coloring, a MIS can be computed in c rounds. • Since rings are bounded independence graphs, algorithm is asymptotically tight. Discrete Algs for Mobile Wireless Sys