40. What is A2[i,j] counts the number of common neighbors
of i and j.
41. What is A2[i,j] counts the number of common neighbors
of i and j.
What is Ak[i,j]?
42. 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 k
between i and j.
44. We know how to count walks!
Suppose we now want to count paths.
45. We know how to count walks!
Suppose we now want to count paths.
Let U be the collection of all Hamiltonian Walks.
46. We know how to count walks!
Suppose we now want to count paths.
Let U be the collection of all Hamiltonian Walks.
Let Ai be the set of walks of length n passing through i.
47. Consider…
n
X=
Ai
i=1
Let U be the collection of all Hamiltonian Walks.
Let Ai be the set of walks of length n passing through i.
48. Consider…
n
X=U
Ai
i=1
Let U be the collection of all Hamiltonian Walks.
Let Ai be the set of walks of length n passing through i.
49. Consider…
n
X=U
Ai
i=1
Let U be the collection of all Hamiltonian Walks.
Let Ai be the set of walks of length n passing through i.
50. Consider…
n
X=U
Ai
i=1
Let U be the collection of all Hamiltonian Walks.
Let Ai be the set of walks of length n passing through i.
The set of walks that avoid i.
51. Consider…
n
X=U
Ai
i=1
|Z|
|X| = |U|
( 1)
Z [n]
Ai
i Z
Let U be the collection of all Hamiltonian Walks.
Let Ai be the set of walks of length n passing through i.
The set of walks that avoid i.
52. Consider…
n
X=U
Ai
i=1
|Z|
|X| = |U|
( 1)
Z [n]
Ai
i Z
Let U be the collection of all Hamiltonian Walks.
Let Ai be the set of walks of length n passing through i.
The set of walks that avoid i.
53. Consider…
n
X=U
Ai
i=1
|Z|
|X| = |U|
( 1)
Z [n]
Ai
i Z
Let U be the collection of all Hamiltonian Walks.
Let Ai be the set of walks of length n passing through i.
The set of walks that avoid i.
The set of walks that avoid X.
54. Consider…
n
X=U
Ai
i=1
|Z|
|X| = |U|
( 1)
Z [n]
Ai
i Z
Let U be the collection of all Hamiltonian Walks.
Let Ai be the set of walks of length n passing through i.
The set of walks that avoid i.
The set of walks that avoid X.
56. 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 polynomials
are equal, finding a perfect matching, primality testing,
and so on.
57. Basic Idea
(x1 + 3x2
x3 )(3x1 + x4
1) · · · (x7
x2 )
Simplifying is expensive, but evaluating is cheap.
0?
58. Basic Idea
(x1 + 3x2
x3 )(3x1 + x4
1) · · · (x7
x2 )
Simplifying is expensive, but evaluating is cheap.
The probability that a random assignment
corresponds to a root is low.
0?
59. 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.
75. For a walk W, we will dump all these terms into the formula:
76. 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)]
77. 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:
78. 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, )
W:=v1 ,...,vk
79.
80. This polynomial captures exactly the paths in G,
i.e, it is identically zero precisely when G has no paths of length k.
