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

Discrete Algs for Mobile Wireless Sys2 Lecture 26  Topic: Maximal Independent Set  Sources: Luby Schneider & Wattenhofer Linial MIT Fall 2008 slides

Discrete Algs for Mobile Wireless Sys3 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 Sys4 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 Sys5 Unit Disk Graphs R R Wireless networks often modeled as unit disk graphs

Discrete Algs for Mobile Wireless Sys6 More Realistic Graphs

Discrete Algs for Mobile Wireless Sys7 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 Sys8 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 Sys9 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(r 2 ).

Discrete Algs for Mobile Wireless Sys10 Maximal vs. Maximum IS a maximum independent seta maximal independent set

Discrete Algs for Mobile Wireless Sys11 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 Sys12 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 Sys13 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 Sys14 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 Sys15 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 Sys16 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 Sys17 Competitions  Every competitor v starts a competition with a number r v and computes new r v ’ initially r v = ID(v)  Computation of r v ’: u: neighboring competitor with minimal r u if r u > r v then r v ’ = 0 else, r v ’ is computed from the base-2 representations of r v and r u : r v ’ is position of highest bit that is 1 in r v and 0 in r u (position of least significant bit is 1)

Discrete Algs for Mobile Wireless Sys18 Competitions  r v ’ is position of highest bit that is 1 in r v and 0 in r u position of least significant bit is 1  Examples: r v = ( ) 2, r u = ( ) 2  r v ’ = 6 r v = ( ) 2, r u = ( ) 2  r v ’ = 4

Discrete Algs for Mobile Wireless Sys19 Competition: New State  Compute new r v ’ based on r v and min r u among neighboring competitors  Update state based on new values of v and neighbors: If r v ’ < r u ’ for all neighboring competitors  v becomes dominator Else if neighbor of v becomes dominator  v becomes dominated Else if r v ’ · r u ’ 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 Sys20 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 Sys21 Reducing the Competitors  Lemma: After log*n + 2 competitions, no node is a competitor any more.  Proof: Initially, r v = ID(v), hence, r v uses at most log n bits Hence, r v ’ uses at most log log n bits After log*n + 2 competitions, r v is in {0,1} All nodes v with r v =0 become dominator or ruler Neighbors become dominated or ruled If r v =1 and all neighboring competitors u have r u =1, v becomes ruler

Discrete Algs for Mobile Wireless Sys22 Phase  log*n+2 competitions are called a phase  For next phase: All rulers become competitors again All r v are set back to ID(v)

Discrete Algs for Mobile Wireless Sys23 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 Sys24 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 Sys25 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 Sys26 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 Sys27 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 Sys28 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.