Today. Continue markov chain mixing analysis. Today. Continue - PowerPoint PPT Presentation

Analyzing random walks on graph. Start at vertex, go to random neighbor. For d -regular graph: eventually uniform. if not bipartite. Odd /even step! How to analyse? Random Walk Matrix: M . M - normalized adjacency matrix. Symmetric, ∑ j M [ i , j ] = 1. M [ i , j ] - probability of going to j from i . Probability distribution at time t : v t . v t + 1 = Mv t Each node is average over neighbors. Evolution? Random walk starts at 1, distribution e 1 = [ 1 , 0 ,..., 0 ] . M t v 1 = 1 N v 1 + ∑ i > 1 λ t i α i v i . v 1 = [ 1 N ,..., 1 N ] → Uniform distribution. Doh! What if bipartite? Negative eigenvalues of value -1: (+ 1 , − 1 ) on two sides. Side question: Why the same size? Assumed regular graph.

Fix-it-up chappie! “Lazy” random walk:

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex.

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] .

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ]

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform:

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i |

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time.

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time. When is chain rapidly mixing?

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time. When is chain rapidly mixing? Another measure:

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time. When is chain rapidly mixing? Another measure: d 2 ( v t , π ) = ∑ i (( v t ) i − π i ) 2 .

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time. When is chain rapidly mixing? Another measure: d 2 ( v t , π ) = ∑ i (( v t ) i − π i ) 2 . √ Note: d 1 ( v t , π ) ≤ Nd 2 ( v t , π )

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time. When is chain rapidly mixing? Another measure: d 2 ( v t , π ) = ∑ i (( v t ) i − π i ) 2 . √ Note: d 1 ( v t , π ) ≤ Nd 2 ( v t , π ) n – “size” of vertex,

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time. When is chain rapidly mixing? Another measure: d 2 ( v t , π ) = ∑ i (( v t ) i − π i ) 2 . √ Note: d 1 ( v t , π ) ≤ Nd 2 ( v t , π ) 1 n – “size” of vertex, µ ≥ p ( n ) for poly p ( n ) ,

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time. When is chain rapidly mixing? Another measure: d 2 ( v t , π ) = ∑ i (( v t ) i − π i ) 2 . √ Note: d 1 ( v t , π ) ≤ Nd 2 ( v t , π ) 1 n – “size” of vertex, µ ≥ p ( n ) for poly p ( n ) , t = O ( p ( n ) log N ) .

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time. When is chain rapidly mixing? Another measure: d 2 ( v t , π ) = ∑ i (( v t ) i − π i ) 2 . √ Note: d 1 ( v t , π ) ≤ Nd 2 ( v t , π ) 1 n – “size” of vertex, µ ≥ p ( n ) for poly p ( n ) , t = O ( p ( n ) log N ) . � 2 t � d 2 ( v t , π ) = | A t e 1 − π | 2 ≤ ( 1 + λ 2 ) 2 p ( n ) ) 2 t ≤ 1 1 ≤ ( 1 − 2 poly ( N )

Fix-it-up chappie! “Lazy” random walk: With probability 1 / 2 stay at current vertex. Evolution Matrix: I + M 2 Eigenvalues: 1 + λ i 2 2 ( v i + λ i v i ) = 1 + λ i 1 2 ( I + M ) v i = 1 2 v i Eigenvalues in interval [ 0 , 1 ] . Spectral gap: 1 − λ 2 = µ 2 . 2 Uniform distribution: π = [ 1 N ,..., 1 N ] Distance to uniform: d 1 ( v t , π ) = ∑ i | ( v t ) i − π i | “Rapidly mixing”: d 1 ( v t , π ) ≤ ε in poly ( log N , log 1 ε ) time. When is chain rapidly mixing? Another measure: d 2 ( v t , π ) = ∑ i (( v t ) i − π i ) 2 . √ Note: d 1 ( v t , π ) ≤ Nd 2 ( v t , π ) 1 n – “size” of vertex, µ ≥ p ( n ) for poly p ( n ) , t = O ( p ( n ) log N ) . � 2 t � d 2 ( v t , π ) = | A t e 1 − π | 2 ≤ ( 1 + λ 2 ) 2 p ( n ) ) 2 t ≤ 1 1 ≤ ( 1 − 2 poly ( N ) 1 Rapidly mixing with big ( ≥ p ( n ) ) spectral gap.

Rapid mixing, volume, and surface area.. Recall volume of convex body.

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body.

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger:

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ .

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time.

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality.

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality. Vol n − 1 ( S , S ) ≥ min ( Vol ( S ) , Vol ( S )) diam ( P )

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality. Vol n − 1 ( S , S ) ≥ min ( Vol ( S ) , Vol ( S )) diam ( P )

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality. Vol n − 1 ( S , S ) ≥ min ( Vol ( S ) , Vol ( S )) diam ( P )

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality. Vol n − 1 ( S , S ) ≥ min ( Vol ( S ) , Vol ( S )) diam ( P ) Edges ∝ surface area,

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality. Vol n − 1 ( S , S ) ≥ min ( Vol ( S ) , Vol ( S )) diam ( P ) Edges ∝ surface area, Assume Diam ( P ) ≤ p ′ ( n )

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality. Vol n − 1 ( S , S ) ≥ min ( Vol ( S ) , Vol ( S )) diam ( P ) Edges ∝ surface area, Assume Diam ( P ) ≤ p ′ ( n ) → h ( G ) ≥ 1 / p ′ ( n )

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality. Vol n − 1 ( S , S ) ≥ min ( Vol ( S ) , Vol ( S )) diam ( P ) Edges ∝ surface area, Assume Diam ( P ) ≤ p ′ ( n ) → h ( G ) ≥ 1 / p ′ ( n ) → µ > 1 / 2 p ′ ( n ) 2

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality. Vol n − 1 ( S , S ) ≥ min ( Vol ( S ) , Vol ( S )) diam ( P ) Edges ∝ surface area, Assume Diam ( P ) ≤ p ′ ( n ) → h ( G ) ≥ 1 / p ′ ( n ) → µ > 1 / 2 p ′ ( n ) 2 → O ( p ′ ( n ) 2 log N ) convergence for Markov chain on BIG GRAPH.

Rapid mixing, volume, and surface area.. Recall volume of convex body. Grid graph on grid points inside convex body. Recall Cheeger: µ � 2 ≤ h ( G ) ≤ 2 µ . Lower bound expansion → lower bounds on spectral gap µ → Upper bound mixing time. h ( G ) ≈ Surface Area Volume Isoperimetric inequality. Vol n − 1 ( S , S ) ≥ min ( Vol ( S ) , Vol ( S )) diam ( P ) Edges ∝ surface area, Assume Diam ( P ) ≤ p ′ ( n ) → h ( G ) ≥ 1 / p ′ ( n ) → µ > 1 / 2 p ′ ( n ) 2 → O ( p ′ ( n ) 2 log N ) convergence for Markov chain on BIG GRAPH. → Rapidly mixing chain:

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n .

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders.

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering.

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order.

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain?

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube.

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube.

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ”

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces.

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces. each of volume: 1 n ! .

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces. each of volume: 1 n ! . since each total order is disjoint

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces. each of volume: 1 n ! . since each total order is disjoint and together cover cube.

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces. each of volume: 1 n ! . since each total order is disjoint and together cover cube. ( 0 , 0 )

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces. each of volume: 1 n ! . since each total order is disjoint and together cover cube. x 1 > x 2 ( 0 , 0 )

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces. each of volume: 1 n ! . since each total order is disjoint and together cover cube. x 1 > x 2 x 1 > x 2 ( 0 , 0 )

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces. each of volume: 1 n ! . since each total order is disjoint and together cover cube. x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 )

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces. each of volume: 1 n ! . since each total order is disjoint and together cover cube. x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3

Khachiyan’s algorithm for counting partial orders. Given partial order on x 1 ,..., x n . Sample from uniform distribution over total orders. Start at an ordering. Swap random pair and go if consistent with partial order. Rapidly mixing chain? Map into d -dimensional unit cube. x i < x j corresponds to halfspace (one side of hyperplane) of cube. “dimension i = dimension j ” total order is intersection of n halfspaces. each of volume: 1 n ! . since each total order is disjoint and together cover cube. x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3

( 0 , 0 )

x 1 > x 2 ( 0 , 0 )

x 1 > x 2 x 1 > x 2 ( 0 , 0 )

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 )

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume.

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume. Number of orders ≡ volume of intersection of partial order relations.

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume. Number of orders ≡ volume of intersection of partial order relations. Diameter: O ( √ n )

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume. Number of orders ≡ volume of intersection of partial order relations. Diameter: O ( √ n ) Isoperimetry:

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume. Number of orders ≡ volume of intersection of partial order relations. Diameter: O ( √ n ) Isoperimetry: Vol n − 1 ( S , S ) = E ( S , S ) | S | ( n − 1 )! ≥ n ! √ n

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume. Number of orders ≡ volume of intersection of partial order relations. Diameter: O ( √ n ) Isoperimetry: Vol n − 1 ( S , S ) = E ( S , S ) | S | ( n − 1 )! ≥ n ! √ n Edge Expansion: the degree d is O ( n 2 ) ,

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume. Number of orders ≡ volume of intersection of partial order relations. Diameter: O ( √ n ) Isoperimetry: Vol n − 1 ( S , S ) = E ( S , S ) | S | ( n − 1 )! ≥ n ! √ n Edge Expansion: the degree d is O ( n 2 ) , h ( S ) = | E ( S , S ) | 1 ≥ d | S | n 7 / 2

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume. Number of orders ≡ volume of intersection of partial order relations. Diameter: O ( √ n ) Isoperimetry: Vol n − 1 ( S , S ) = E ( S , S ) | S | ( n − 1 )! ≥ n ! √ n Edge Expansion: the degree d is O ( n 2 ) , n 7 / 2 Mixes in time O ( n 7 log N ) h ( S ) = | E ( S , S ) | 1 ≥ d | S |

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume. Number of orders ≡ volume of intersection of partial order relations. Diameter: O ( √ n ) Isoperimetry: Vol n − 1 ( S , S ) = E ( S , S ) | S | ( n − 1 )! ≥ n ! √ n Edge Expansion: the degree d is O ( n 2 ) , n 7 / 2 Mixes in time O ( n 7 log N ) = O ( n 8 log n ) . h ( S ) = | E ( S , S ) | 1 ≥ d | S |

x 1 > x 2 x 1 > x 3 x 1 > x 2 ( 0 , 0 ) x 2 > x 3 Each order takes 1 n ! volume. Number of orders ≡ volume of intersection of partial order relations. Diameter: O ( √ n ) Isoperimetry: Vol n − 1 ( S , S ) = E ( S , S ) | S | ( n − 1 )! ≥ n ! √ n Edge Expansion: the degree d is O ( n 2 ) , n 7 / 2 Mixes in time O ( n 7 log N ) = O ( n 8 log n ) . h ( S ) = | E ( S , S ) | 1 ≥ d | S | Do the polynomial dance!!!