Presentation Martin Randers

Transcription

1 Presenttion Mrtin Rnders

2 Outline Introduction Algorithms Implementtion nd experiments Memory consumption Summry

3 Introduction

4 Introduction Evolution of species cn e modelled in trees Trees consist of nodes nd edges nodes divided into leves nd internl nodes edges connecting internl nodes re clled internl edges

5 The Qurtet Distnce A qurtet is set of four leves Cn hve four possile qurtet topologies c c d c d d c d Butterfly topologies Str topology The qurtet distnce etween pir of trees is the numer of qurtets with different topologies

6 Other Algorithms Previously: Binry trees lgorithms creted y Tsng O n 2 & Brodl et l. O n log n Now: Trees of ritrry degrees Generliztion of Tsngs lgorithm uses time O V d 2 V ' d ' 2 Generliztion of Brodl et l.'s lgorithm to trees of ounded degree uses time O d 9 n log n

7 Algorithms

8 Algorithms Wys to compute shred lef set sizes re presented in thesis Five lgorithms for computing the qurtet distnce designed nd implemented: Two center sed Two sed on edge climing One sed on node climing

9 Center Bsed Algorithms Oservtion: Ech triplet of leves hs unique internl node s center

10 Center Bsed Algorithms Oservtion: Ech triplet of leves hs unique internl node s center The center of, nd c c

11 Center Bsed Algorithms Oservtion: Ech triplet of leves hs unique internl node s center The center of, nd c c The topologies of qurtets contining the triplet esily determined from this center

12 Center Bsed Algorithms Oservtion: Ech triplet of leves hs unique internl node s center The center of, nd c c The topologies of qurtets contining the triplet esily determined from this center y x c

13 Center Bsed Algorithms Oservtion: Ech triplet of leves hs unique internl node s center The center of, nd c c The topologies of qurtets contining the triplet esily determined from this center y c x x y The topologies c c

14 First Center Bsed Algorithm O n 4 lgorithm: For ech triplet of leves: Find center of the triplet, in ech tree Determine topology of qurtets, in ech tree Compre topology of qurtets Uses spce O(n) to store topologies

15 Second Center Bsed Algorithm O n 3 lgorithm: Precompute shred lef set sizes For ech pir of leves: Find center of ech triplet contining the pir, in ech tree Use precomputed sizes to clculte numer of shred qurtets Uses spce to store shred lef set sizes O n 2

16 Hndling Str Qurtets Qurtets cn e divided into five ctegories: Q SS T,T ',Q SB T,T ',Q BS T,T ', Q B=B T,T ',Q B B T,T '

17 Hndling Str Qurtets Qurtets cn e divided into five ctegories: Q SS T,T ',Q SB T,T ',Q BS T,T ', Q B=B T,T ',Q B B T,T ' qdist T,T '

18 Hndling Str Qurtets Qurtets cn e divided into five ctegories: Q SS T,T ',Q SB T,T ',Q BS T,T ', Q B=B T,T ',Q B B T,T ' qdist T,T ' Oserve tht Q B=B T,T =Q BS T,T ' Q B=B T,T ' Q B B T,T ' Q B=B T ',T ' =Q SB T,T ' Q B=B T,T ' Q B B T,T '

19 Hndling Str Qurtets Qurtets cn e divided into five ctegories: Oserve tht Yielding Q SS T,T ',Q SB T,T ',Q BS T,T ', Q B=B T,T ',Q B B T,T ' qdist T,T ' Q B=B T,T =Q BS T,T ' Q B=B T,T ' Q B B T,T ' Q B=B T ',T ' =Q SB T,T ' Q B=B T,T ' Q B B T,T ' qdist T,T ' =Q B=B T,T Q B=B T ',T ' 2Q B=B T,T ' Q B B T,T '

20 Climing Associting qurtets to specific edges or nodes in the trees Clculte shred/nonshred qurtets y processing pirs of edges/nodes Importnt tht ech qurtet is climed y fixed numer of edges/nodes

21 Edge Climing Ech directed internl edge e clims ll utterfly qurtets where two leves re locted in two different sutrees in front of e, nd the other two re in the sutree ehind e e x c d

22 Edge Climing Ech directed internl edge e clims ll utterfly qurtets where two leves re locted in two different sutrees in front of e, nd the other two re in the sutree ehind e e x c d e clims cx nd dx, ut not cd

23 Edge Climing Ech directed internl edge e clims ll utterfly qurtets where two leves re locted in two different sutrees in front of e, nd the other two re in the sutree ehind e e x c d e clims cx nd dx, ut not cd Ech utterfly qurtet is climed y exctly two directed edges

24 Encoding The qurtets climed y ech edge cn e encoded s numer of triplets of sutrees (clims) The numer of shred nd nonshred qurtets in pir of clims cn e computed in time O(1) If e points to node of degree d, it tkes d 1 clims to encode the qurtets climed Processing ll pirs of clims tkes time 2 v V v' V ' id v d v 1 2 id v ' d v' 1 2 =O V V ' d 2 d ' 2

25 Expnsion First edge climing lgorithm: Expnd the trees to inry trees, nd define extended clims Compute shred lef set sizes Process ll pirs of extended clims If e points to node of degree d, it tkes d 1 extended clims to encode the qurtets climed Processing ll pirs of extended clims thus tkes time v V v' V ' id v d v id v ' d v' =O V V ' dd ' Uses spce O n 2 to store shred lef set sizes

26 Counting Insted of encoding, count numer of shred/nonshred qurtets climed y oth edges directly Done in three steps for pir of edges: i. Count too mny (some illegl ) ii. Deduct ll the illegl ones (some twice) iii. Add the ones tht were deducted twice If pir of edges points to nodes of internl degrees id nd id' respectively this cn e done in time O(idid')

27 Counting (cont) Second edge climing lgorithm: Compute shred lef set sizes of non lef sutrees Do the counting descried ove for ech pir of edges Doing precomputtions nd processing ll pirs of edges tkes time O n V V ' v V v ' V ' id 2 2 v id =O n v ' V V ' idid ' Uses spce O n V V ' sizes nd sizes of sutrees to store shred lef set

28 Node Climing Motivtion: A lot of redundnt computtions re done in the second edge climing lgorithm Ech internl node v clims ll utterfly qurtets climed y directed internl edges pointing to v Ech utterfly qurtet is climed y exctly two directed edges

29 Shred Qurtets Given pir of nodes v nd v' numer of sums cn e precomputed in time O id v id v' O id v id v ' nd spce These enles the numer of shred qurtets climed y oth nodes to e computed in time O id v id v '

30 Nonshred Qurtets By precomputing n dditionl sums O min {id v,id v ' } 2 in time O id v id v ' min {id v,id } v' it is lso possile to compute the numer of nonshred qurtets climed y the nodes in time O id v id v ' min {id v,id v ' }

31 The Algorithm Node climing lgorithm: Compute shred lef set sizes of non lef sutrees For ech pir of nodes: Do the precomputing descried ove Compute shred nd nonshred qurtets This tkes time O n V V ' min {id,id ' }... nd uses spce O n V V '

32 Implementtion & Experiments

33 Implementtion All of the presented lgorithms hve een implemented in Jv Tool, with optionl GUI, ville for downlod Experiments hs een performed using these implementtions

34 Experiments The correctness of lgorithms vlidted y compring results The lgorithms were run on four clsses of trees to investigte ctul running times: worst cse d ry rndom r8s sed

35 Experiments The correctness of lgorithms vlidted y compring results The lgorithms were run on four clsses of trees to investigte ctul running times: worst cse d ry rndom r8s sed

36 Exmple of Results

37 Compring the Algorithms

38 More Comprison

39 Anlysis of Results The symptotic performnce ws s expected The second edge climing nd the node climing re clerly fster thn the others. The node climing lgorithm ws chosen for the tool

40 Reducing Memory Consumption

41 Reducing Memory Consumption The work hs een focused t reducing running time, not memory consumption Reduction in memory from O n 2 to ws just yproduct of optimizing running time Cn further reduction e done, mye t the cost of dding running time? O n V V '

42 Importnt Fcts When processing pir of nodes it is necessry nd sufficient to know the shred lef set sizes of ll pirs of non lef sutrees of the nodes, nd the sizes of individul sutrees Implies tht trivil lower ound on memory consumption is O(n+id id') The chllenge:

43 Importnt Fcts When processing pir of nodes it is necessry nd sufficient to know the shred lef set sizes of ll pirs of non lef sutrees of the nodes, nd the sizes of individul sutrees Implies tht trivil lower ound on memory consumption is O(n+id id') The chllenge: The shred lef set sizes of sutrees re clculted using other shred lef set sizes, i.e. they re not independent

44 Bsic Ide The ide is inspired y the O(n log(n)) lgorithm:

45 Bsic Ide The ide is inspired y the O(n log(n)) lgorithm: Do numer of colorings of the leves in one tree

46 Bsic Ide The ide is inspired y the O(n log(n)) lgorithm: Do numer of colorings of the leves in one tree Build some structure sed on the other tree nd updte it when colors chnge

47 Coloring Let v e n internl node in tree T. T is sid to e colored ccording to v if the leves of ech non lef sutree of v re colored with the colors 1,...,id v, one for ech sutree, nd ll leves directly connected to v re colored with the color 0

48 Coloring Let v e n internl node in tree T. T is sid to e colored ccording to v if the leves of ech non lef sutree of v re colored with the colors 1,...,id v, one for ech sutree, nd ll leves directly connected to v re colored with the color 0 c v f g d e

49 Coloring Let v e n internl node in tree T. T is sid to e colored ccording to v if the leves of ech non lef sutree of v re colored with the colors 1,...,id v, one for ech sutree, nd ll leves directly connected to v re colored with the color 0 c f v color g v c f g 0,1,2,3 d e d e

50 Structure Root the other tree in n ritrry internl node, nd dd n rry to ech internl node with id v entries Color the leves in the rooted tree Initilize the entries in the rrys with the numer of leves of ech color directly connected to the node Updte the rrys depth first with the sum of the entries in the rrys of the children

51 Exmple d c r g e f

52 Exmple d c r g root r e f d c e f g

53 Exmple d c r g root r e f dd rrys d c e f g r g d c e f

54 Exmple d c r g root r e f dd rrys d c e f g r r g color leves g 0,1,2,3 d c e f d c e f

55 Exmple d c e f r g root dd rrys d c e f r g r r g color leves g 0,1,2,3 d c e f r g init rrys d c e f d c e f

56 Exmple d c r e f d c e f g r root dd rrys g color leves r init rrys g updte rrys d c e f d c e f r d c e f d c e f g r r g g 0,1,2,3

57 Anlysis Rooting cn e done once nd for ll in constnt time For ech node v in T: Coloring cn e done in time O(n) Adding rrys tkes time Updting rrys tkes time Spce consumption is O V ' id v O V ' id v O v ' V ' id v id v' =O V ' id v

58 Anlysis (cont) Computtion cn e done s efore, except the nodes must e hndled in n ordered wy The time to hndle ll pirs of nodes is O V V ' min {id,id ' } Totl complexities re thus: O V n V V ' min {id,id ' } time, nd O V ' id spce Note tht the roles of the trees cn e switched

59 Summry

60 Overview Presented here: The prolem Existing lgorithms A wy to void str qurtets The five lgorithms designed nd implemented

61 Overview Presented here: The prolem Existing lgorithms A wy to void str qurtets The five lgorithms designed nd implemented

62 Overview Presented here: The prolem Existing lgorithms A wy to void str qurtets The five lgorithms designed nd implemented Some of the experiments performed A wy to reduce the memory consumption of the fstest lgorithm (t performnce cost)

63 Additionl Work in Thesis Additionl work in the thesis: Wys to compute shred lef set sizes Generliztion of existing mesures Hndling non stndrd input trees Visuliztion Presenttion of tool Comprison with split distnce

64 The End