1. 1 Approximation Algorithms for Low- Distortion Embeddings into Low- Dimensional Spaces Badoiu et al. (SODA 2005) Presented by: Ethan Phelps-Goodman Atri Rudra

2. 2 Moving from Seattle to NY Looks like a 2hr drive  Contraction is bad Looks like a 10hr plane ride  Expansion is bad “Faithful” representation

3. 3 Metric Spaces Set of points Distance function  Non-negative  Triangle Inequality Metric  Graph  weighted graph  shortest path distance 2 1 3 5 5 6

4. 4 Embeddings Mapping  : X ! Y Exp(  )=max a,b 2 X d 2 (  (a),  (b))/d 1 (a,b) Contr(  )=Exp(  -1 ) Dist(  )=Exp(  ) with Contr(  ) ¸ 1 (X,d 1 ( ¢, ¢ )) (Y,d 2 ( ¢, ¢ ))

5. 5 Edge property Mapping from a unwt graph G=(V,E) Exp(  )=max (x,y) 2 E d 2 (x’,y’)  Exp(  )=max x 2 V,y 2 V d 2 (x’,y’)/d 1 (x,y)  ¸ is obvious  x=v 1,v 2, ,v L =y be the shortest path in G max i d 2 (v’ i,v’ i+1 ) ¸ 1/L ¢  i d 2 (v’ i,v’ i+1 ) The sum ¸ d 2 (x’,y’) by triangle inequality

6. 6 Embedding into a class of metrics Pick an embedding Non-contractive Minimum distortion Embedding into specific Metric [KRS04],[PS05]

7. 7 Previous work Start with a problem in some metric space Embed inputs into another “nice” metric  Problem is easy to solve in the nice metric  For e.g., embed graphs into trees Combinatorial in nature  Try to upper bound the distortion Survey by Indyk D

8. 8 Algorithmic Task Find embedding with min possible distortion  Don’t care about the value of distortion  Recall the map example Embed unweighted graph into R (line) as well as possible

9. 9 What is in store for you ? Embed unwt graph ! Line Hardness of the algorithmic Q Approx Algo for general unwt graph Approx Algo for unwt tree What else is there in the paper ? Open problems

10. 10 Hardness of Approximation NP-Hard to a-approximate for some a>1 Reduction from TSP on (1,2)-metric  All distances are in {1,2}  NP-Hard to a-approximate for any a <5381/5380 (X,D) ) G=(X,E)  D(u,v)=1  (u,v) 2 E

11. 11 Hardness of Approximation TSP on (1,2)-metric  Input M:- (V,D( ¢, ¢ ))  Output:- Permutation of V which is a tour  M  G=(V,E) G x yD(x,y)=1  (x,y) 2 E

12. 12 The Construction Given metric M  G Make a copy of G called G’=(V’,E’) Add a special node o connected to V [ V’ The constrcuted graph H has diameter=2 G

13. 13 Showing that it works M has a tour of len t ) H embeds into R with distr · t Let the tour be v 1,v 2, ,v n,v 1  Start with v 1  Lay out v 2, ,v n according to their distances  Put in o  Lay out G’ in the same way as G R v1v1 0 v2v2 vivi v i+1 vnvn D(v i,v i+1 ) o 1 v’ 1 v’ n 1 Non-contractive Expansion=t

14. 14 The other direction f embeds H into R w/ distr s ) 9 tour in M of len · s+1 Let u 1, ,u 2n be the ordering of V [ V’ by f  Assume “nice” ordering  |f(u 2n )-f(u 1 )| · 2s  (Wlog) blue box · (2s-2)/2=s-1  Blue box gives a tour of length · (s-1)+2=s+1 R f(u 1 ) f(u 2n ) ¸ 1

15. 15 Getting a nice ordering Total ‘span’ still · 2s “Boundary” nodes are at distance ¸ 2 Swap blue and green boxes  Total Span still · 2s  Still non-contractive Keep on swapping till nice ordering is reached R ¸ 2

16. 16 What is in store for you ? Embed unwt graph ! Line Hardness of the algorithmic Q Approx Algo for general unwt graph Approx Algo for unwt tree What else is there in the paper ? Open problems

17. 17 Embedding a tree into R Create an Eulerian tour Embed according to tour  preserve order  preserve distances No contraction Distortion · 2n-1

18. 18 Embedding general graphs Every graph embeds into R w/ O(n) distortion  Use any spanning tree Coming Soon: c-approx algo  c is the value of the optimal distortion Combining both gives O(n 1/2 )-approx  If c · n 1/2, c-approx algo works  If c> n 1/2, spanning tree algo works

19. 19 c-approx algorithm Embed G=(V,E) into R Let f * be the optimal embedding Let t 1,t 2 be the ‘end-points’ of f * t 1 =v 1,v 2, ,v L =t 2 shortest path V=V 1 [ V 2  [ V L  x closest to v i ) x 2 V i

20. 20 c-approx algo (contd.) Embed V i (i=1,  L) by the spanning tree algo  Layout v i first  Recall that the max span is · 2|V i | Leave a gap of |V i | on each side Run for all possible values of t 1 and t 2 V1V1 VLVL V2V2 V L-1 ViVi R |V i | 2|V i |

21. 21 Notations f * is the optimal embedding c is the optimal distortion f is the computed embedding D( ¢, ¢ ) shortest path in G

22. 22 Analysis Clm1: D(v i,x) · c/2 Clm2: |V i |+|V i+1 |+  +|V i+c-1 | · 2c 2 Clm3: Embedding is non-contracting Will now show |f(x)-f(y)| · 16c 2  |i-j| · 2c  Span= 4 ¢ [ (|V i |+  +|V i+c-1 |) + (|V i+c |+  +|V j |) ] The constructed embedding has distortion O(c 2 ) V1V1 VLVL V2V2 V L-1 ViVi R |V i | 2|V i | x VjVj y 4|V i | as D(v i,v j ) · D(x,v i )+D(x,y)+D(y,v+j) · c/2+1+c/2 f * has distortion c

23. 23 Non-contractive embedding x,y 2 V i x 2 V i, y 2 V j  |f(x)-f(y)| ¸ |V i | + 2(|V i+1 |+  |V j-1 |) +|V j | ¸ |V i | + |j-i| + |V j | ¸ D(x,v i ) + D(v i,v j ) + D(v j,y) ¸ D(x,y) V1V1 VLVL V2V2 V L-1 ViVi R |V i | 2|V i | x VjVj y y Should be |V j |+1 (root goes first)

24. 24 Proof of Clm1 2 ¢ D(x,v i ) · ? c  D(x,v i ) + D(x,v i ) · D(x,v j )+D(x,v j+1 ) · ( f * (v j ) – f * (x) ) + ( f * (x) – f * (v j+1 ) = f * (v j ) – f * (v j+1 ) · c · c R x vivi vjvj V j+1 f*(v i ) f*(v j ) f*(v j+1 )

25. 25 Embedding Trees into the line Problem: Given an unweighted tree that embeds into the line with distortion c, find the smallest distortion line embedding. They give (c logc) 1/2 -approximation, Can also be stated as O((n logn) 1/3 )- approximation:  If c > n 2/3 then use simple spanning tree algorithm  If c < n 2/3 then use this algorithm

26. 26 Tree Embeddings Similar to previous algorithm Select endpoints & compute shortest path

27. 27 Tree Embeddings Similar to previous algorithm Select endpoints & compute shortest path Group every c vertices Embed each component, then concatenate

28. 28 Local Density Define local density by   = max v, r (|B(v, r)|-1)/2r. Then c > .  In max density ball, there are 2r  vertices, so end points of embedding have distance at least 2r .  But max distance is 2r, so endpoints have distortion at least . Also, any component with diameter d has at most d  vertices.

29. 29 Component Embedding Note: only care about order of vertices. Distances computed from shortest path Want to embed in roughly depth-first order But don’t want neighbors too far away Algorithm alternates between laying out neighbors of previously visited vertices (BFS) and DFS.

30. 30 Component Embedding in action CiCi

31. 31 Component embedding algorithm Magic number g(c)= 2  (clogc) 1/2 + c Pick a leftmost vertex r Let C i be vertices visited up to round i While there are unvisited vertices  Visit all neighbors of C i  Visit next g(c) vertices in light-path DFS order

32. 32 Bounding the distortion Outline:  Bound number of iterations  Bound span of neighbor step  Bound total distortion in component  Bound distortion from concatenation

33. 33 Number of iterations Diameter of tree is at most 2c: So total # vertices is 2c  At least g(c) added at each iteration Number of iterations is (2c  )/g(c)  (clog -1 c) 1/2 c c/2

34. 34 Distortion of neighbor set or, where did g(c) come from? Claim: neighbor set is spanned by tree of size g(c). Idea: Vertices in neighbor set can’t be too far from “active” DFS vertices:  at most (i+1) < (clog -1 c) 1/2 away. So spanned by tree of size  (clog -1 c) 1/2 2c 2 vertices in component, so 2logc can be active 2logc *  (clog -1 c) 1/2 + c = g(c).

35. 35 Distortion for full component Vertices added in neighbor step are spanned by tree of size g(c) g(c) connected vertices added in DFS step So distortion at most 2g(c) for each iteration 2 adjacent vertices could be on opposite ends of iteration i and i+1, so total distortion 4g(c) over all iterations

36. 36 Concatenating the embeddings There is only one edge (v i, v i+1 ) connecting components X i and X i+1 Modify the DFS ordering of X i so that v i is last visited Doesn’t affect distortion of X i, and distortion of edge (v i, v i+1 ) is at most 2g(c) Total distortion is at most  4g(c) = 8  (clogc) 1/2 + 4c  8 c 3/2 log 1/2 c + 4c  Or O((c logc) 1/2 ) times optimal

37. 37 Other algorithms in paper An exact algorithm for embedding a general graph into the line, with runtime O(n c ). A geometric 3-approximation for embedding the sphere into the plane.

38. 38 Open questions For lines:  Better approximation ratios  Lower bounds  Weighted graphs (with large distortion) Bigger question: algorithmic embeddings of graphs into the plane.