Building Optimal Websites with the Constrained Subtree Selection Problem

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Building Optimal Websites with the Constrained Subtree Selection Problem

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Title: Building Optimal Websites with the Constrained Subtree Selection Problem

1
Building Optimal Websites with the Constrained
Subtree Selection Problem

Brent Heeringa
(joint work with Micah Adler)
09 November 2004

2
A website design problem(for example a new
kitchen store)

Given products, their popularity, and their
organization
How do we create a good website?
Navigation is natural
Access to information is timely

3
Good website Natural Navigation

Organization is a DAG
TC of DAG enumerates all viable categorical
relationships and introduces shortcuts
Subgraph of TC preserves logical relationship
between categories

TC
4
Good website Timely Access to Info

Two obstacles to finding info quickly
Time scanning a page for correct link
Time descending the DAG
Associate a cost with each obstacle
Page cost (function of out-degree of node)
Path cost (sum of page costs on path)
Good access structure
Minimize expected path cost
Optimal subgraph is always a full tree

1/2
Page Cost links Path Cost 325 Weighted
Path Cost 5/2
5
Constrained Subtree Selection (CSS)

An instance of CSS is a triple (G,?,w)
G is a rooted, DAG with n leaves (constraint
graph)
? is a function of the out-degree of each
internal node (degree cost)
w is a probability distribution over the n
leaves (weights)
A solution is any directed subtree of the
transitive closure of G which includes the root
and leaves
An optimal solution is one which minimizes the
expected path cost

1/4
1/4
1/4
1/4
?(x)x
6
Constrained Subtree Selection (CSS)

An instance of CSS is a triple (G,?,w)
G is a rooted, DAG with n leaves (constraint
graph)
? is a function of the out-degree of each
internal node (degree cost)
w is a probability distribution over the n
leaves (weights)
A solution is any directed subtree of the
transitive closure of G which includes the root
and leaves
An optimal solution is one which minimizes the
expected path cost

1/4
1/4
1/4
1/4
3(1/4)
5(1/4)
5(1/4)
3(1/4)
?(x)x Cost4
7
Constrained Subtree Selection (CSS)

An instance of CSS is a triple (G,?,w)
G is a rooted, DAG with n leaves (constraint
graph)
? is a function of the out-degree of each
internal node (degree cost)
w is a probability distribution over the n
leaves (weights)
A solution is any directed subtree of the
transitive closure of G which includes the root
and leaves
An optimal solution is one which minimizes the
expected path cost

1/2
1/6
1/6
1/6
?(x)x Cost 3 1/2
8
Constraint-Free Graphs and k-favorability

Constraint-Free Graph
Every directed, full tree with n leaves is a
subtree of the TC
CSS is no longer constrained by the graph
k-favorable degree cost ?
Fix ?. There exists kgt1 for any constraint-free
instance of CSS under ? where an optimal tree
has maximal out-degree k

9
Linear Degree Cost - ?(x)x

5 paths w/ cost 5

3 paths w/ cost 5
2 paths w/ cost 4

10
Linear Degree Cost - ?(x)x
gt 1/2

Prefer binary structure when a leaf has at least
half the mass

Prefer ternary structure when mass is
uniformly distributed

CSS with 2-favorable degree costs and C.F.
graphs is Huffman coding problem
Examples quadratic, exp, ceiling of log

11
Results

Complexity NP-Complete for equal weights and
many ?
Sufficient condition on ?
Hardness depends on constraint graph
Highlighted Results
Theorem O(n?(k)k)-time DP algorithm
? is integer-valued, k-favorable and G is
constraint free
?(x)x
Theorem poly-time constant-approximation
?1 and k-favorable G has constant out-degree
Approximate Hotlink Assignment - Kranakis et.
al
Other results
Characterizations of optimal trees for uniform
probability distributions

12
Related Work

Adaptive Websites Perkowitz Etzioni
Challenge to the AI community
Novel views of websites Page synthesis problem
Hotlink Assignment Kranakis, Krizanc, Shende,
et. al.
Add 1 hotlink per page to minimize expected
distance from root to leaves
Recently pages have cost proportional to their
size
Hotlinks dont change page cost
Optimal Prefix-Free Codes Golin Rote
Min code for n words with r symbols where symbol
ai has cost ci
Resembles CSS without a constraint graph

13
Exact Cover by 3-Sets
INPUT (X,C) X(x1,,xn) n3k and C(C1,,Cm) Ci
? X OUTPUT C ? C where Ck and covers X
QUESTION Given K and (X,C) is there a cover of
size K?
Sufficient condition on ? For every integer
k, there exists an integer s(k) such that
14
(X,C) X(x1,,xn) n3k and C(C1,,Cm) Ci ? X
15
Lopsided Trees

Recall ?(x)x, and G is constraint free
Node level path cost
Adding an edge increases level
Grow lopsided trees level by level

16
Lopsided Trees
17
Lopsided Trees
18
Lopsided Trees
19
Lopsided Trees

We know exact cost of tree up to the current
level i
Exact cost of m leaves
Remaining n-m leaves must have path-cost at
least i

20
Lopsided Trees

Exact cost of C 3 (1/3)1
Remaining mass up to level 4 (2/3) 4 8/3
Total 18/311/3

21
Lopsided Trees

Tree cost at Level 5 in terms of Tree cost at
Level 4
Add in the mass of remaining leaves
Cost at Level 5
No new leaves
11/32/313/3

22
Lopsided Trees
23
Lopsided Trees
24
Lopsided Trees

Equality on trees
Equal number of leaves at or above frontier
Equal number of leaves at each relative level
below frontier
Nodes have outdegree 3
Node below frontier ?(3)
(ml1, l2, l3) signature
Example Signature (2 3, 2, 0)
2 C and F are leaves
3 G, H, I are 1 level past the frontier
2 J and K are 2 levels past the frontier

25
Inductive Definition

Let CSS(m,l1,l2,l3) min cost tree with sig.
(ml1, l2, l3)
Can we define CSS(m,l1,l2,l3) in terms of optimal
substructures?
Which trees, when grown by one level, have
signatures CSS(m,l1,l2,l3)?
Which signatures (m,l1,l2,l3) lead to
(m,l1,l2,l3)

26
The other direction
Sig (0 2, 0, 0)

Growing a tree only affects frontier
Only l1 affects next level
Choose leaves
The remaining nodes are internal
Choose degree-2 (d2)
Remaining nodes are degree-3 (d3)
O(n2) choices

Sig (1 0, 0, 3)
27
The original question(warning here be symbols)

Which (ml1,l2,l3) (ml1,l2,l3)
l1 and d2 are sufficient
l1 and d2 are both O(n)
O(n2) possibilities for (ml1,l2,l3)
CSS(m,l1,l2,l3) min cost tree with sig. (ml1,
l2, l3)
CSS(m,l1,l2,l3)
cm for 1d2l1n
(cm are the smallest n-m weights)
CSS(n,0,0,0) cost of optimal tree
Analysis
Table size O(n4)
Each cell takes O(n2) lookups
O(n6) algorithm

28
Lower Bound on Cost

Lemma H(w)/log(k) is a lower bound on the cost
of an optimal tree
For any k-favorable degree cost ?, with ?1
G is constraint-free

T
T
T
1
1
1
1
1
1
1
1
1
c(T) c(T)
c(T) H(w)/log(k)
(shannon)
29
A Simple Lemma

Lemma 2 For any tree with m weighted nodes
there exists 1 node (splitter) which, when
removed, divides the tree into subtrees with at
most half the weight of the original tree.

splitter
lt1/2
lt 1/2
lt 1/2
30
Aproximation Algorithm

Let G be a DAG where out-degree of every node ? d
Choose a spanning tree T from G
Balance-Tree(T)
Find a splitter node in T (Lemma 2)
Stop if splitter is child of root
Disconnect the splitter and reconnect it to the
root
root has degree at most d1
Call Balance-Tree on all subtrees

splitter
Mass of each subtree is at least half of whole
tree
31
Approximation Algorithm

Analysis
Mass under any node is half of mass under its
grandparent
Path length to leaf with weight wi is -2log(wi)
Theorem
O(m)-time O(log(k)?(d1))-approx to optimal
solution
For any DAG G with m nodes and out-degree ? d
For every k-favorable degree cost ? 1,

Upper Bound on Node Cost
Weighted Path Length
32
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33
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34
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35
Proposed Problem 1(CSS in constraint-free
graphs, equal leaf weights)

Question Polytime algorithm for CSS with
Constraint-free graphs
Equal leaf weights
Increasing degree cost
Good News
Characterizations for linear and log degree costs
Near linear time algorithms for r-ary Varn Codes
(Huffman codes with r unequal letter costs,
uniform probability distribution)

36
Varn Codes(infinite lopsided tree)
Symbol Costs (3,3,3,8,8)
5 Leaves
Note Not the 5 highest Leaves!
37
Varn Codes(infinite lopsided tree)
Symbol Costs (3,3,3,8,8)
6 Leaves
Note m internal nodes are the highest m nodes
in the infinite tree
38
Proposed Problem 1(CSS in constraint-free
graphs, equal leaf weights)

Bad News
No Notion of an infinite lopsided tree in CSS
Degree change structure change
Optimal CSS tree is fairly balanced
Property
No leaf may appear above the level of any other
internal node
Proof If it were the case, we could switch
branches and decrease the cost of the tree
Intuition There is some k which optimizes
breadth-to-depth tradeoff. The optimal tree
repeats this structure. Fringe requires some
computation time.

39
Proposed Problem 2(Dynamic CSS)

CSS often applies to environments which are
inherently dynamic
Web pages change popularity
Access patterns change on file systems
Question Given a CSS tree with property P, how
much time does it take to maintain P after an
update?
P minimum cost, approximation-ratio of min cost
Restrict attention to
Integer leaf weights (rational distributions)
Unit updates

40
Proposed Problem 2(Dynamic CSS)

Good News Knuth (and later Vitter) studied
Dynamic Huffman Codes (DHC)
Motivation One-pass encoding
Protocol
Both parties maintain optimal tree for first t
characters
Encode and decode t1 character
Update tree
Optimality of tree maintained in time
proportional to encoding

41
DHC Sibling Property

A binary tree with n leaves is a Huffman tree
iff
The n leaves have nonnegative weights w1wn
the weight of each internal node is the sum of
the weights of its children
The nodes can be numbered in non-decreasing order
by weight
siblings are numbered consecutively
common parent has a higher number

32
11
10
21
Numbering corresponds to merging in greedy
algorithm
11
9
F
11
10
7
8
5
4
5
6
5
3
5
6
D
E
C
2
3
1
2
A
B
42
DHC Sibling Property

A binary tree with n leaves is a Huffman tree
iff
The n leaves have nonnegative weights w1wn
the weight of each internal node is the sum of
the weights of its children
The nodes can be numbered in non-decreasing order
by weight
siblings are numbered consecutively
common parent has a higher number

33
11
10
22
What happens if we increase B? Node 4 violates
the Sibling Property
11
9
F
11
11
7
8
6
4
5
6
5
3
5
6
D
E
C
2
4
1
2
A
B
43
DHC Sibling Property

A binary tree with n leaves is a Huffman tree
iff
The n leaves have nonnegative weights w1wn
the weight of each internal node is the sum of
the weights of its children
The nodes can be numbered in non-decreasing order
by weight
siblings are numbered consecutively
common parent has a higher number

32
11
10
21
Before updating Exchange current node with node
with highest number having the same weight
11
9
F
11
10
7
8
5
4
5
6
5
3
5
6
D
E
C
2
3
1
2
A
B
44
DHC Sibling Property

A binary tree with n leaves is a Huffman tree
iff
The n leaves have nonnegative weights w1wn
the weight of each internal node is the sum of
the weights of its children
The nodes can be numbered in non-decreasing order
by weight
siblings are numbered consecutively
common parent has a higher number

32
11
10
21
Before updating Exchange current node with node
with highest number having the same weight
11
9
F
11
10
7
8
5
4
5
6
5
3
5
6
D
E
C
2
3
1
2
A
B
45
DHC Sibling Property

A binary tree with n leaves is a Huffman tree
iff
The n leaves have nonnegative weights w1wn
the weight of each internal node is the sum of
the weights of its children
The nodes can be numbered in non-decreasing order
by weight
siblings are numbered consecutively
common parent has a higher number

32
11
10
21
Different, but still optimal, greedy choice when
merging nodes
11
9
F
11
10
7
8
4
5
6
5
3
5
6
5
D
E
C
2
1
2
3
A
B
46
DHC Sibling Property

A binary tree with n leaves is a Huffman tree
iff
The n leaves have nonnegative weights w1wn
the weight of each internal node is the sum of
the weights of its children
The nodes can be numbered in non-decreasing order
by weight
siblings are numbered consecutively
common parent has a higher number

32
11
10
21
Different, but still optimal, greedy choice when
merging nodes
11
9
F
11
10
7
8
4
5
6
5
3
5
6
5
D
E
C
2
1
2
3
A
B
47
DHC Sibling Property

A binary tree with n leaves is a Huffman tree
iff
The n leaves have nonnegative weights w1wn
the weight of each internal node is the sum of
the weights of its children
The nodes can be numbered in non-decreasing order
by weight
siblings are numbered consecutively
common parent has a higher number

33
11
10
21
Different, but still optimal, greedy choice when
merging nodes
11
9
7
10
11
8
6
5
6
5
F
E
4
5
5
3
3
2
2
C
D
1
A
B
48
DHC Sibling Property

A binary tree with n leaves is a Huffman tree
iff
The n leaves have nonnegative weights w1wn
the weight of each internal node is the sum of
the weights of its children
The nodes can be numbered in non-decreasing order
by weight
siblings are numbered consecutively
common parent has a higher number

32
11
10
21
Now, safe to increase B, because it cant be
greater than the next highest!
11
9
7
10
11
8
6
5
6
5
F
E
4
5
5
3
3
2
2
C
D
1
A
B
49
DHC Sibling Property

A binary tree with n leaves is a Huffman tree
iff
The n leaves have nonnegative weights w1wn
the weight of each internal node is the sum of
the weights of its children
The nodes can be numbered in non-decreasing order
by weight
siblings are numbered consecutively
common parent has a higher number

33
11
10
21
Now, safe to increase B, because it cant be
greater than the next highest!
12
9
7
10
11
8
6
5
6
6
F
E
4
5
5
3
4
2
2
C
D
1
A
B
50
Proposed Problem 2(Dynamic CSS)

Good News DHC generalizes to k-ary alphabets
Claim
DHC is an O(?(k))-approximation for CSS
? k-favorable, ?(x)1
constraint-free graphs

51
Proposed Problem 2(Dynamic CSS)

Bad News DHC doesnt generalize to Huffman
codes with unequal letter costs
Sibling property Greedy algorithm
Future
Explore DHC for unequal letter costs
Maintain approximation ratio in constant degree
graphs in time proportional to the height
(We can do it in linear time already)

52
Proposed Problem 3(Category Tree - CT)

Scenario
Large reservoir of songs in iTunes
Song is a vector of categorical values
Common to search all the songs for the right one
Question Can we organize the songs by
categories so that the average search time is
minimized?

53
Proposed Problem 3(Category Tree - CT)

Category Tree CT(?,C,S)
? is the degree cost
C(d1,,dm) are the m category sizes
S is a set of objects drawn from C
Solution Rooted, oriented tree
Internal nodes are categories
Edges are appropriate categorical values
Leaves are objects
Optimal solution
Minimize expected path cost
Path cost is defined as in CSS

Optimal solution corresponds to an adaptive
ordering of the categories
54
Proposed Problem 3(Constrained Category Tree -
CCT)

Constrained Category Tree CCT(?,C,S)
? is the degree cost
C(d1,,dm) are the m category sizes
S is a set of objects drawn from C
Solution Rooted, oriented tree
Internal nodes are categories (and internal nodes
at the same depth have the same category)
Edges are appropriate categorical values
Leaves are objects
Optimal solution
Minimize expected path cost
Path cost is defined as in CSS

Optimal solution corresponds to a fixed ordering
of the categories
55
Proposed Problem 3(Category Tree - CT)

CT and CCT are classical Decision Tree problems

Decision Tree (DT) Input m binary tests
T(T1Tm) and n objects O(O1On) Output Binary
tree where internal nodes are Ti and leaves or
Oi Measure Total external path length

CT and CCT are NP-Complete
Reduction from Exact Cover by 3-Sets (XC3)
Resembles hardness proof for Decision Tree

56
Proposed Problem 3(Category Tree - CT)
Decision Tree Inference (DTI) Input m
examples T/F labeled binary strings from 0,1n
Output Binary tree where internal nodes are
string positions and leaves are TRUE or FALSE
which is consistent with examples Measure Number
of leaves (i.e. size of tree)

CT and CCT are not instances of DTI
DT doesnt easily reduce to DTI
Most complexity results (lower bounds on
approximations) are for DTI only!

57
Timetable

Solve some subset of open problems
1-2 academic years

58
Open Problems

Theorem There is an for any instance (G,?,w) of
CSS where G is constraint free, ? is
k-favorable, maps the positive integers to the
positive integers and is non-decreasing

Proof
c(T) c(T) c(T) H(w)/log(k)
T is optimal tree for CSS cost c
T is optimal tree for OPC cost c for k symbols
each with weight 1 (i.e. ?(x)1)
H is entropy

59
Signatures as Representation

Different lopsided trees share common
substructure when truncated
Level-i-Truncations Include node iff parent is
at most i
Level-i-Signatures ml1,..,l?(k)
m is the of leaves level i
lj is of nodes at level ij
Cost of Level-i-Truncation
Exact cost for m leaves
Cost up to the truncation for the remaining n-m
leaves.

60
The Dynamic Programming Table

Signatures Table entries
MINml1,..,l?(k) gives min-cost of all
truncated trees with signature ml1,..,l?(k)
O(n?(k)1) entries
level-i truncation is parent of O(nk-1)
level-(i1) truncation
level-i sig is parent of
O(nk-1) level-(i1) sigs
Choose how many nodes at next level will be
internal
Among those, choose how many will be degree 2,
degree 3, , degree k O(nk-1) choices
Consistent ordering of entries
O(n?(k)k) algorithm MINn0,,0 contains
minimum cost

Set of products
The desired information
e.g., chef paring knives
Popularity of products
Weights
Hierarchical organization of products into
categories
Single, global category (the root)
Products are endpoints (leaves)
General to specific trajectory

Adaptive Websites Perkowitz Etzioni
Page synthesis (novel view) with clustering and
concept learning using access logs
Efficiently find topic of interest (effort)
Hotlink Assignment Kranakis, Krizanc, Shende,
et. al.
Add k hotlinks per page to minimize expected
distance from root to leaves
Recently pages have fixed cost proportional to
their size
Hotlinks dont change path-cost
Optimal Prefix-Free Codes Golin Rote
Min code for n words with r symbols where symbol
ai has cost ci
Resembles CSS without a constraint graph

63
Lopsided Trees

ml1,..,l?(k) MINm,l1,..,l?(k)
n leaves so at most O(n?(k)1) entries
Entry stores minimum cost of tree bearing that
signature
Total ordering on signatures, consistent with the
growing process
O(nk-1) choices
O(n?(k)k) algorithm

64
Lopsided Trees

Tree cost at Level 5 in terms of Tree cost at
Level 4
Cost at Level 5 11/32/313/3
Cost at Level 6 13/31/229/6

65
The original question(warning here be symbols)

Which (m,l1,l2,l3) (m,l1,l2,l3)

66
The original question(warning here be symbols)

Which (m,l1,l2,l3) (m,l1,l2,l3)
Suppose we know
l1 (the of nodes one level below the frontier)
d2 (the of l1 which are degree-2 nodes in
(m,l1,l2,l3))
Lets determine the values of the remaining
variables

1
1
2
2
3
d2 nodes
l1 nodes
3
67
The original question(warning here be symbols)

Which (m,l1,l2,l3) (m,l1,l2,l3)
Suppose we know
l1 (the of nodes one level below the frontier)
d2 (the of l1 which are degree-2 nodes in
(m,l1,l2,l3))

The old number of leaves
Internal nodes of degree 2
m m l1 - d2 - d3
Nodes at one level below the frontier
Internal nodes of degree 3
The new number of leaves
68
The original question(warning here be symbols)

Which (m,l1,l2,l3) (m,l1,l2,l3)
Suppose we know
l1 (the of nodes one level below the frontier)
d2 (the of l1 which are degree-2 nodes in
(m,l1,l2,l3))

The old number of leaves
Internal nodes of degree 2
m m l1 - d2 - l3/3
Nodes at one level below the frontier
Internal nodes of degree 3
The new number of leaves
69
The original question(warning here be symbols)