The Power of Locality
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1 GIAN Course on Distributed Network Algorithms The Power of Locality Case Study: Graph Coloring Stefan T-Labs, 2011
2 Case Study: Graph Coloring
3 Case Study: Graph Coloring Assign colors to nodes.
4 Case Study: Graph Coloring Legal coloring: neighbors have different colors!
5 Case Study: Graph Coloring Optimal coloring: Minimal number of colors (aka chromatic number)
6 Applications Country Maps Medium Access Image Processing Neighboring states should have different colors! Famous 4-color theorem: any map can be painted with four colors! Interference-free, efficient utilization of spectrum Neighboring cells should have different frequencies! Colors = frequencies, channels, etc. Chromatic scheduling for physical simulation Process nodes of same color in parallel without determinacy race No coordination, no mutual exclusion needed 6
7 Legal coloring? Chromatic number? 7
8 Legal coloring? Chromatic number? Tree! 2 colors are enough 8
9 What about this example? 9
10 What about this example? 3 colors needed and enough 10
11 Now distributed! How to color a graph in a distributed manner? Stefan T-Labs,
12 Now distributed! How to color a graph in a distributed manner? Stefan T-Labs,
13 The LOCAL Model: A Convenient Synchronous Model in each Send round! receive compute. 13
14 The LOCAL Model: A Convenient Synchronous Model We will see in this course: there are techniques to execute an algorithm designed in the simple LOCAL model also in asynchronous networks! in each Send round! receive compute. 14
15 We will see in this course: there are The LOCAL Model: A techniques Convenient to execute Synchronous an algorithm Model designed in the simple LOCAL model also in asynchronous networks! in each Send round! Moreover, LOCAL algorithms can be made very robust (namely selfstabilizing), in an automatic manner! receive compute. 15
16 We will see in this course: there are The LOCAL Model: A techniques Convenient to execute Synchronous an algorithm Model designed in the simple LOCAL model also in asynchronous networks! in each Send round! Moreover, LOCAL algorithms can be made very robust (namely selfstabilizing), in an automatic manner! receive Unlike CONGEST model: message size and link capacity not bounded. compute. 16
17 LOCAL Performance Metrics Time Complexity: Number of communication rounds Message Complexity: Number of messages sent Local Computation: Complexity of local computations
18 LOCAL Performance Metrics What else? Time Complexity: Number of communication rounds Message Complexity: Number of messages sent Local Computation: Complexity of local computations
19 LOCAL Performance Metrics What else? Time Complexity: Quality of solution: Approximation ratio for example ( price of locality ). Number of communication rounds Message Complexity: Number of messages sent Local Computation: Complexity of local computations
20 Now distributed!how to color a tree in a distributed manner? Stefan T-Labs, 2011
21 Now distributed! How to color a rooted tree in a distributed manner? Simplification: Assume unique node IDs Assume rooted Root ID 0 0 Stefan T-Labs, 2011
22 Now distributed! How to color a rooted tree in a distributed manner? Simplification: Assume unique node IDs Assume rooted Root ID 0 0 Idea: interpret ID as color! Communicate my color to children and take opposite color from my parent! Stefan T-Labs, 2011
23 Slow Distributed Tree Coloring: Example 0 Round 1 23
24 Slow Distributed Tree Coloring: Example Round 2 24
25 Slow Distributed Tree Coloring: Example Round 3 25
26 Slow Distributed Tree Coloring: Example Round
27 Slow Tree Slow Tree Algo If root: color 0, send 0 to children Otherwise: each node v: Wait for message x from parent Choose color y=1-x Send y to children Stefan T-Labs,
28 Slow Tree: Analysis Approximation quality: Time complexity: Message complexity: Local complexity: 28
29 Slow Tree: Analysis Approximation quality: # colors? Time complexity: # rounds? Message complexity: # messages? Local complexity: local computations? 29
30 Slow Tree: Analysis Approximation quality: 2 colors suffice! Time complexity: O(n), depth of the tree Message complexity: O(n) Local complexity: trivial, just flip! 30
31 Slow Tree: Analysis Can we do faster? Approximation quality: 2 colors suffice! Time complexity: O(n), depth of the tree Message complexity: O(n) Local complexity: trivial, just flip! 31
32 Ultra Fast Distributed Tree Coloring Yes we can! 3-coloring in O(log* n) rounds 32
33 Ultra Fast Distributed Tree Coloring Yes we can! 3-coloring in O(log* n) rounds Idea: based on ID manipulations Again: interpret ID as color Unique IDs legal (but expensive) coloring! How can we quickly reduce the ID space? 33
34 Intuition: n vs log* n log n: How many times do I have to :2 until <2? n, n/2, n/4, n/8,, 8, 4, 2, 1 log n
35 Intuition: n vs log* n log n: How many times do I have to :2 until <2? n, n/2, n/4, n/8,, 8, 4, 2, 1 log n loglog n: How many times do I have to x until <2? n, n, n, n,, <2 loglog n
36 Intuition: n vs log* n log n: How many times do I have to :2 until <2? n, n/2, n/4, n/8,, 8, 4, 2, 1 log n loglog n: How many times do I have to x until <2? n, n, n, n,, <2 loglog n log* n: How many times do I have to log x until <2? n, log n, loglog n, logloglog n,, <2 log* n
37 n = atoms in universe log*(atoms in universe) 5
38 Slow Algo No parallelism! Fast Algo Efficient parallel manipulations! Time: n Time: log* n
39 Log*-Time Coloring with Label Manipulation Initially ID = label of node v =color c v initially 39
40 Log*-Time Coloring with Label Manipulation Initially ID = label of node v =color c v log n Log n bits to represent n unique IDs initially Initially ID = label of node v =color c v Unique IDs legal (but expensive) coloring! 40
41 Log*-Time Coloring with Label Manipulation Send ID to children Round Send ID to children Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 41
42 Log*-Time Coloring with Label Manipulation Round Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) 42 ID = color for next round: the position!
43 Log*-Time Coloring with Label Manipulation Example: Differ at position 5 = (0101) Differ at position 8 = (1000) Round Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) 43
44 Log*-Time Coloring with Label Manipulation Differ at position 5 = (0101) Differ at position 8 = (1000) How long are the new IDs? Round 1 Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) 44
45 Log*-Time Coloring with Label Manipulation Differ at position 5 = (0101) Differ at position 8 = (1000) How long are the new IDs? Round 1 Describing position in x-bit string takes log x bits, so: loglog n bits Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) 45
46 Log*-Time Coloring with Label Manipulation Differ at position 5 = (0101) Differ at position 8 = (1000) How long are the new IDs? Round 1 Describing position in x-bit string takes log x bits, so: loglog n bits Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) +1 bit 46
47 Log*-Time Coloring with Label Manipulation Round Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) 47
48 Log*-Time Coloring with Label Manipulation Differ at position 3 = (11) Round Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) 48
49 Log*-Time Coloring with Label Manipulation How long are the new IDs? Differ at position 3 = (11) Round Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) 49
50 Log*-Time Coloring with Label Manipulation How long are the new IDs? Differ at position 3 = (11) Round Describing position in x-bit string takes log x bits, so: logloglog n bits Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) +1 bit 50
51 Log*-Time Coloring with Label Maniplulation 111 Round Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) 51
52 Log*-Time Coloring with Label Maniplulation 111 etc.! 001 Algorithm: in round i, node v: 1. Send my c v to children (in parallel!) 2. Receive parent ID/color c p 3. Let i be the smallest index where c v and c p differ (from right, binary) 4. My new c v = i c v (i) 52
53 Analysis How long does it take until O(1) colors? Why is coloring always legal? 53
54 Analysis How long does it take until O(1) colors? # bits/colors reduced by a log-factor in each round The definition of log*! log* n: How many times do I have to log x until <2? Why is coloring always legal? Algorithm: My new c v = i c v (i) 54
55 Analysis How long does it take until O(1) colors? # bits/colors reduced by a log-factor in each round The definition of log*! log* n: How many times do I have to log x until <2? Why is coloring always legal? Algorithm: My new c v = i c v (i) grandfather: c g By contradiction: To get the same ID as my father, I need to differ at same position from father as father from grandfather. But then last bit must be different: there I took my own bit (and father will do the same with his different bit)! father: c f me: c v 55
56 Summary of Algorithm 56
57 Summary of Algorithm Note: we stop if color in {0,,5}: why? 57
58 Summary of Algorithm Note: we stop if color in {0,,5}: why? Could I go down to 2-bit colors, i.e., {0,,3}: No, requires 2 bits to address index where they differ, plus adding the difference-bit gives more than two bits. 58
59 Summary of Algorithm Note: We we stop if color in {0,,5}: why? Could I I go down to to2-bit colors, i.e., i.e., {0,,3}: No, requires 2 bits 2 bits to to address index where they differ, plus adding the the difference-bit gives more than two bits. For 3-bit colors {0,,7} this still works: e.g., 7=(111) 2 can be described with 3 bits, and position index (0,1,2) requires two bits, plus one difference-bit gives three again 59
60 Summary of Algorithm Note: We we stop if color in {0,,5}: why? Could I I go down to to2-bit colors, i.e., i.e., {0,,3}: No, requires 2 bits 2 bits to to address index where they differ, plus adding the the difference-bit gives more than two bits. For 3-bit colors {0,,7} this still works: e.g., 7=(111) 2 can be described with 3 bits, and position index (0,1,2) requires two bits, plus one difference-bit gives three again But actually colors 110 (for color 6 ) and 111 (for color 7 ) are not needed, as we can do another round! IDs of three bits can only differ at positions 00 (for 0 ), 01 (for 1 ), 10 (for 2 ) 60
61 With 6-COLORS algorithm we can get down to 6 colors. What about improving it to 2 colors?
62 With 6-COLORS algorithm we can get down to 6 colors. What about improving it to 2 colors? Impossible: takes linear time. What about 3 colors?
63 Observation: Shift Down Let us note a simple trick: shift colors down by one level makes siblings independent. And preserves legal coloring
64 Observation: Shift Down Let us note a simple trick: shift colors down by one level makes siblings independent. And preserves legal coloring Result: all my descendants have same color! At most 2 colors are occupied: father and descendants! 3rd color free!
65 Observation: Shift Down Let us note a simple trick: shift colors down by one level makes siblings independent. And preserves legal coloring Shift Down Each node v concurrently does: recolor v with color of parent Formally Result: all my descendants have same color! At most 2 colors are occupied: father and descendants! 3rd color free!
66 6-to-3 6-to-3 Each other node v does (in parallel): 1. Run 6-Colors for log*(n) rounds 2. For x=5,4,3: 1. Perform Shift Down 2. If (c v =x) choose new color c v ϵ {0,1,2} according first free principle 66
67 6-to-3 6-to-3 Each other node v does (in parallel): 1. Run 6-Colors for log*(n) rounds 2. For x=5,4,3: 1. Perform Shift Down 2. If (c v =x) choose new color c v ϵ {0,1,2} according first free principle Why still log* n time? 67
68 6-to-3 6-to-3 Each other node v does (in parallel): 1. Run 6-Colors for log*(n) rounds 2. For x=5,4,3: 1. Perform Shift Down 2. If (c v =x) choose new color c v ϵ {0,1,2} according first free principle Why still log* n time? Just 3 more rounds! 68
69 6-to-3 6-to-3 Why not do in same step? Each other node v does (in parallel): 1. Run 6-Colors for log*(n) rounds 2. For x=5,4,3: 1. Perform Shift Down 2. If (c v =x) choose new color c v ϵ {0,1,2} according first free principle 69
70 6-to-3 6-to-3 Why not do in same step? Each other node v does (in parallel): 1. Run 6-Colors for log*(n) rounds 2. For x=5,4,3: 1. Perform Shift Down 2. If (c v =x) choose new color c v ϵ {0,1,2} according first free principle Could be harmful: same 3rd color! Need to do it for E.g. 5 => 1 independent sets. E.g. 4 => 1
71 Example: Shift Down + Drop Color 4 4 shift down! shift down 1 first free for 4! Siblings no longer have same color: must do shift down again first! 71
72 Example: 6-to-3 4 shift down new color for 5: first free 2 shift down Careful: cannot recolor 4 at same time!
73 Remark: Optimality One can show that no local algorithm can 3-color a graph faster than in O(log* n). 73
74 Remark: Optimality One can show that no local algorithm can 3-color a graph faster than in O(log* n). In fact: in 0 rounds: n colors in 1 round: log n colors in 2 rounds: loglog n colors etc.! 74
75 Remark: Optimality One can show that no local algorithm can 3-color a graph faster than in O(log* n). In fact: in 0 rounds: n colors in 1 round: log n colors in 2 rounds: loglog n colors etc.! Proof idea: Recall the elephant! A local coloring algorithm can be seen as a function: f: neighborhood color A deterministic algorithm needs to decide in the same way given same neighborhood: risk illegal coloring. Only with communication neighborhoods start look different and require less colors. 75
76 Lower Bound Set of neighborhoods Local coloring algo Vertex coloring 4 ALG ALG
77 Lower Bound Can reduce problem of finding lower bound to determine chromatic number of special neighborhood dependency graphs. Set of neighborhoods Local coloring algo Vertex coloring 4 ALG ALG
78 Concluding Remarks Can we reduce to 2 colors? Not without increasing runtime significantly! (Linear time, more than exponentially worse!) Simple on purpose: results more general! log* runtime is also possible on more general graphs Many results: see ACM PODC conference! 78
79 Where can I learn more? Distributed Computing book by David Peleg Lecture notes by Roger Wattenhofer ETH Zurich ACM Survey by Jukka Suomela Research: ACM PODC Conference 79
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