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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balanced Programming
Group Static
Shanghai Jiao Tong University
December 27, 2016
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Overview
1 Introduction
2 Baby step giant step
3 Another Example
4 Meet in the middle
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Features
Kind of a useful designing idea.
Easy implementation and good performance in practice.
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Features
Kind of a useful designing idea.Easy implementation and good performance in practice.
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balances
Balanced in learning from each each other.
Balanced in the problem-solving processes.
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balances
Balanced in learning from each each other.Balanced in the problem-solving processes.
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Available occiasion
There are two algorithms, both of which have differentfeatures, such as one can answer queries very quickly and theother cam handle modification swiftly.
There is an algorithm to solve the problems, but is quitecomplicated. Actually this algorithm is not the best choice inpractice. The idea of balanced programming might be used toproduced a suitable algorithm.
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Available occiasion
There are two algorithms, both of which have differentfeatures, such as one can answer queries very quickly and theother cam handle modification swiftly.There is an algorithm to solve the problems, but is quitecomplicated. Actually this algorithm is not the best choice inpractice. The idea of balanced programming might be used toproduced a suitable algorithm.
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Overview
1 Introduction
2 Baby step giant step
3 Another Example
4 Meet in the middle
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Problem
Find a smallest non-negative number x satisfying
Ax ≡ B (mod P)
which P is a prime, A,B ∈ [0,P).
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach
According to Fermat Theorem, when A,P are coprime, we have:
AP ≡ A (mod P)
the only special case is A = 0, and in this case, B must be zero.
For A ̸= 0, we can just iterate x from 0 to P− 1 to check ifAx ≡ B (mod P). The complexity is O(P).
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Yet Another Naive Algorithm
We mapped Ax → x, x ∈ [0,P) by using a Hash-Table.
In order to find B, we just need to query B in this Hash-Table,which only takes O(1) time.
Precalculating this Hash-Table costs O(P) time, and the totalcomplexity of which is O(P + 1) = O(P).
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balanced Programming
First we choose a number S ∈ [1,P− 1].We mapped Ax → x, x ∈ [0,S) by using a Hash-Table.Calculate A−S by using Fast Exponentiation.
In this step, we needs S + log P operations.
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balanced Programming
Ax ≡ B (mod P)
x can be represented as i× S + j, we can do such transforming:
Ai×S+j ≡ B⇔ Aj ≡ B× (A−S)i (mod P)
which j ∈ [0,S), and i < PS .
We can just iterate i from 0 to PS to check if B× (A−S)i is in
Hash-Table. In the worst occasion, we need to iterate PS times.
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Total complexity evaluation
As you can see, the total operations we need to do are
S + log P +PS
we want to choose S optimally to get the best total complexity.
when S =√
P, the total operations are:
S + log P +PS = 2
√P + log P = O(
√P)
thus we get an O(√
P) algorithm by choosing S optimally.
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Overview
1 Introduction
2 Baby step giant step
3 Another Example
4 Meet in the middle
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Another Problem
an undirected graph
add x to uquery neighbors’sumO(m) = n
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Another Problem
an undirected graphadd x to u
query neighbors’sumO(m) = n
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Another Problem
an undirected graphadd x to u
query neighbors’sumO(m) = n
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Another Problem
an undirected graphadd x to u
query neighbors’sumO(m) = n
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Another Problem
an undirected graphadd x to uquery neighbors’sum
O(m) = n
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Another Problem
an undirected graphadd x to uquery neighbors’sum
O(m) = n
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Another Problem
an undirected graphadd x to uquery neighbors’sumO(m) = n
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 1
store the value v[x]
O(n) spaceO(1) time for addO(n) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 1
store the value v[x]O(n) space
O(1) time for addO(n) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 1
store the value v[x]O(n) spaceO(1) time for add
O(n) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 1
store the value v[x]O(n) spaceO(1) time for addO(n) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 2
store the sum s[x]
O(n) spaceO(n) time for addO(1) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 2
store the sum s[x]O(n) space
O(n) time for addO(1) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 2
store the sum s[x]O(n) spaceO(n) time for add
O(1) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 2
store the sum s[x]O(n) spaceO(n) time for add
O(1) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 2
store the sum s[x]O(n) spaceO(n) time for add
O(1) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Naive Approach 2
store the sum s[x]O(n) spaceO(n) time for addO(1) time for query
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Observation
bad when large deg(x)
”heavy” when deg(x) > B
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Observation
bad when large deg(x)”heavy” when deg(x) > B
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Heavy-Light Divide
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Heavy-Light Divide
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Combined Approach
s[x] =∑v[heavy neighbors] +∑v[light neighbors]
for∑
v[heavy neighbors]use approach 1for
∑v[light neighbors]
use approach 2
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Combined Approach
s[x] =∑v[heavy neighbors] +∑v[light neighbors]
for∑
v[heavy neighbors]use approach 1for
∑v[light neighbors]
use approach 2
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Add(u, x) details
if u is heavy,vh[u]← vh[u] + x
if u is light,sl[v]← sl[v] + x,u, v are neighborsO(1) time for heavyO(B) time for light
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Add(u, x) details
if u is heavy,vh[u]← vh[u] + x
if u is light,sl[v]← sl[v] + x,u, v are neighborsO(1) time for heavyO(B) time for light
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Add(u, x) details
if u is heavy,vh[u]← vh[u] + xif u is light,sl[v]← sl[v] + x,u, v are neighbors
O(1) time for heavyO(B) time for light
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Add(u, x) details
if u is heavy,vh[u]← vh[u] + xif u is light,sl[v]← sl[v] + x,u, v are neighborsO(1) time for heavy
O(B) time for light
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Add(u, x) details
if u is heavy,vh[u]← vh[u] + xif u is light,sl[v]← sl[v] + x,u, v are neighborsO(1) time for heavyO(B) time for light
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Query(x) details
∑v[heavy neighbors] =∑y is heavy neighbor vh[y]
∑v[light neighbors] = sl[x]
O(1 + cnt_heavy) timecnt_heavy · B ≤ 2m = O(n)O(n/B) time
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Query(x) details
∑v[heavy neighbors] =∑y is heavy neighbor vh[y]∑v[light neighbors] = sl[x]
O(1 + cnt_heavy) timecnt_heavy · B ≤ 2m = O(n)O(n/B) time
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Query(x) details
∑v[heavy neighbors] =∑y is heavy neighbor vh[y]∑v[light neighbors] = sl[x]
O(1 + cnt_heavy) timecnt_heavy · B ≤ 2m = O(n)O(n/B) time
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Query(x) details
∑v[heavy neighbors] =∑y is heavy neighbor vh[y]∑v[light neighbors] = sl[x]
O(1 + cnt_heavy) time
cnt_heavy · B ≤ 2m = O(n)O(n/B) time
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Query(x) details
∑v[heavy neighbors] =∑y is heavy neighbor vh[y]∑v[light neighbors] = sl[x]
O(1 + cnt_heavy) timecnt_heavy · B ≤ 2m = O(n)
O(n/B) time
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Query(x) details
∑v[heavy neighbors] =∑y is heavy neighbor vh[y]∑v[light neighbors] = sl[x]
O(1 + cnt_heavy) timecnt_heavy · B ≤ 2m = O(n)O(n/B) time
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balance
O(n) space
O(B) time for addO(n/B) time for querymake B =
√n
O(√
n) for each operation
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balance
O(n) spaceO(B) time for add
O(n/B) time for querymake B =
√n
O(√
n) for each operation
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balance
O(n) spaceO(B) time for addO(n/B) time for query
make B =√
nO(√
n) for each operation
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balance
O(n) spaceO(B) time for addO(n/B) time for querymake B =
√n
O(√
n) for each operation
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Balance
O(n) spaceO(B) time for addO(n/B) time for querymake B =
√n
O(√
n) for each operation
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Overview
1 Introduction
2 Baby step giant step
3 Another Example
4 Meet in the middle
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Search algorithm
Search algorithmexponential :O(xdepth)
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Meet in the middle
Given an undirectedgraph whose nodes’degree is 5 and finda path from S to T.
dist(S, T) = LO(4L)
Meet in the middleO(4L/2)
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Meet in the middle
Given an undirectedgraph whose nodes’degree is 5 and finda path from S to T.dist(S, T) = L
O(4L)
Meet in the middleO(4L/2)
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Meet in the middle
Given an undirectedgraph whose nodes’degree is 5 and finda path from S to T.dist(S, T) = LO(4L)
Meet in the middleO(4L/2)
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Meet in the middle
Given an undirectedgraph whose nodes’degree is 5 and finda path from S to T.dist(S, T) = LO(4L)
Meet in the middleO(4L/2)
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Some other applications
Mo’s algorithm
Blocked listDinic(capacity is 1)......
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Some other applications
Mo’s algorithmBlocked list
Dinic(capacity is 1)......
Group Static Balanced Programming
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IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Some other applications
Mo’s algorithmBlocked listDinic(capacity is 1)
......
Group Static Balanced Programming
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Some other applications
Mo’s algorithmBlocked listDinic(capacity is 1)......
Group Static Balanced Programming
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
...
.
IntroductionBaby step giant step
Another ExampleMeet in the middle
Thanks
Thanks
Thanks for listening.Questions are welcomed.
Group Static Balanced Programming
