Download - Paths and Polynomials
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Paths & Polynomials
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Hamiltonian Path has an algorithm thatspends O(2n) time andconsumes O(2n) space.
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Walks to the rescue…
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Walks to the rescue…
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Walks to the rescue…
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
What is A2[i,j]?
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
What is A2[i,j]?
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
What is A2[i,j]?
i
j
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
What is A2[i,j]?
i
j
![Page 36: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/36.jpg)
Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
What is A2[i,j]?
i
j
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
What is A2[i,j]?
i
j
![Page 38: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/38.jpg)
Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
What is A2[i,j]?
i
j
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Adjacency Matrix A[i,j] = 1 iff (i,j) is an edge in G.
What is A2[i,j]?
i
j
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What is A2[i,j] counts the number of common neighbors of i and j.
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What is A2[i,j] counts the number of common neighbors of i and j.
What is Ak[i,j]?
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What is A2[i,j] counts the number of common neighbors of i and j.
What is Ak[i,j]?
What is Ak[i,j] counts the number of walks of length kbetween i and j.
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We know how to count walks!
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We know how to count walks!
Suppose we now want to count paths.
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We know how to count walks!
Suppose we now want to count paths.
Let U be the collection of all Hamiltonian Walks.
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We know how to count walks!
Suppose we now want to count paths.
Let Ai be the set of walks of length n passing through i.
Let U be the collection of all Hamiltonian Walks.
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Let Ai be the set of walks of length n passing through i.
Let U be the collection of all Hamiltonian Walks.
Consider…
X =n�
i=1
Ai
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Let Ai be the set of walks of length n passing through i.
Let U be the collection of all Hamiltonian Walks.
Consider…
X = U �n�
i=1
Ai
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Let Ai be the set of walks of length n passing through i.
Let U be the collection of all Hamiltonian Walks.
Consider…
X = U �n�
i=1
Ai
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Let Ai be the set of walks of length n passing through i.
Let U be the collection of all Hamiltonian Walks.
Consider…
X = U �n�
i=1
Ai
The set of walks that avoid i.
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Let Ai be the set of walks of length n passing through i.
Let U be the collection of all Hamiltonian Walks.
Consider…
X = U �n�
i=1
Ai
The set of walks that avoid i.
|X| = |U| ��
Z�[n]
(�1)|Z|�
i�Z
Ai
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Let Ai be the set of walks of length n passing through i.
Let U be the collection of all Hamiltonian Walks.
Consider…
X = U �n�
i=1
Ai
The set of walks that avoid i.
|X| = |U| ��
Z�[n]
(�1)|Z|�
i�Z
Ai
![Page 53: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/53.jpg)
Let Ai be the set of walks of length n passing through i.
Let U be the collection of all Hamiltonian Walks.
Consider…
X = U �n�
i=1
Ai
The set of walks that avoid i.
|X| = |U| ��
Z�[n]
(�1)|Z|�
i�Z
Ai
The set of walks that avoid X.
![Page 54: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/54.jpg)
Let Ai be the set of walks of length n passing through i.
Let U be the collection of all Hamiltonian Walks.
Consider…
X = U �n�
i=1
Ai
The set of walks that avoid i.
|X| = |U| ��
Z�[n]
(�1)|Z|�
i�Z
Ai
The set of walks that avoid X.
![Page 55: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/55.jpg)
Polynomial Identity Testing
Input: A polynomial p(x). Question: Is p(x) identically zero?
(x1 + 3x2 � x3)(3x1 + x4 � 1) · · · (x7 � x2) � 0?
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Polynomial Identity Testing
Input: A polynomial p(x). Question: Is p(x) identically zero?
(x1 + 3x2 � x3)(3x1 + x4 � 1) · · · (x7 � x2) � 0?
Captures several problems, like checking if two polynomialsare equal, finding a perfect matching, primality testing,
and so on.
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Basic Idea
(x1 + 3x2 � x3)(3x1 + x4 � 1) · · · (x7 � x2) � 0?
Simplifying is expensive, but evaluating is cheap.
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Basic Idea
(x1 + 3x2 � x3)(3x1 + x4 � 1) · · · (x7 � x2) � 0?
Simplifying is expensive, but evaluating is cheap.
The probability that a random assignmentcorresponds to a root is low.
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Given a (directed) graph G and a number k… !!!
Let’s try to construct a polynomial that is zero !
if and only if !
G has a path of length k.
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x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Edges
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Vertices
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x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Edges
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Vertices
We want terms corresponding to walks to cancel out,
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x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Edges
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Vertices
We want terms corresponding to walks to cancel out,
somehow.
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The Trick
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Evaluate these polynomials over finite fieldsof characteristic two.
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Evaluate these polynomials over finite fieldsof characteristic two.
a + a = 0
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x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Edges
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Vertices
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x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Edges
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Vertices
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
![Page 70: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/70.jpg)
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Edges
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Vertices
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
![Page 71: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/71.jpg)
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Edges
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Vertices
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
![Page 72: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/72.jpg)
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Edges
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Vertices
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
![Page 73: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/73.jpg)
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Edges
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
Vertices
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1]y[v2] · · · y[vk]
![Page 74: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/74.jpg)
![Page 75: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/75.jpg)
For a walk W, we will dump all these terms into the formula:
![Page 76: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/76.jpg)
For a walk W, we will dump all these terms into the formula:
�
�:[k]�[k]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1, �(1)]y[v2, �(2)] · · · y[vk, �(k)]
![Page 77: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/77.jpg)
For a walk W, we will dump all these terms into the formula:
�
�:[k]�[k]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1, �(1)]y[v2, �(2)] · · · y[vk, �(k)]
Sum this over all walks W:
![Page 78: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/78.jpg)
For a walk W, we will dump all these terms into the formula:
�
�:[k]�[k]
x[v1,2]x[v2,3] · · · x[vk�1,k]y[v1, �(1)]y[v2, �(2)] · · · y[vk, �(k)]
Sum this over all walks W:
�
W:=v1,...,vk
�(W, �)
![Page 79: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/79.jpg)
![Page 80: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/80.jpg)
This polynomial captures exactly the paths in G,
i.e, it is identically zero precisely when G has no paths of length k.
![Page 81: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/81.jpg)
�
W:=v1,...,vk
�(W, �)
![Page 82: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/82.jpg)
�
W:=v1,...,vk
�(W, �)
How do we evaluate this polynomial now?
![Page 83: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/83.jpg)
�
W:=v1,...,vk
�(W, �)
How do we evaluate this polynomial now?
![Page 84: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/84.jpg)
![Page 85: Paths and Polynomials](https://reader036.vdocument.in/reader036/viewer/2022062419/559137a11a28ab0d498b460a/html5/thumbnails/85.jpg)
Inclusion ExclusionRELOADED