Monotone Paths Po-Shen Loh Carnegie Mellon University Joint work with Mikhail Lavrov
Monotone sequences Theorem (Erd˝ os-Szekeres 1935) Every permutation of { 1 , . . . , n } has a monotone subsequence of length about √ n .
Monotone sequences Theorem (Erd˝ os-Szekeres 1935) Every permutation of { 1 , . . . , n } has a monotone subsequence of length about √ n . Example 1 5 2 7 3 6 4
Monotone sequences Theorem (Erd˝ os-Szekeres 1935) Every permutation of { 1 , . . . , n } has a monotone subsequence of length about √ n . Example 1 5 2 7 3 6 4 Proof. Under each number, write lengths of longest increasing and decreasing subsequences ending there. 1 5 2 7 3 6 4 inc. 1 2 2 3 3 4 4 dec. 1 1 2 1 2 2 3
Monotone walks: lower bound Question (Chv´ atal-Komlos 1971) � n If edges of K n are ordered from 1 . . . � , is there always a long 2 monotone
Monotone walks: lower bound Question (Chv´ atal-Komlos 1971) � n If edges of K n are ordered from 1 . . . � , is there always a long 2 monotone walk?
Monotone walks: lower bound Question (Chv´ atal-Komlos 1971) � n If edges of K n are ordered from 1 . . . � , is there always a long 2 monotone walk? Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing walk of length n − 1.
Monotone walks: lower bound Question (Chv´ atal-Komlos 1971) � n If edges of K n are ordered from 1 . . . � , is there always a long 2 monotone walk? Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing walk of length n − 1. Proof. 1 3 2
Monotone walks: lower bound Question (Chv´ atal-Komlos 1971) � n If edges of K n are ordered from 1 . . . � , is there always a long 2 monotone walk? Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing walk of length n − 1. Proof. 1 3 2
Monotone walks: lower bound Question (Chv´ atal-Komlos 1971) � n If edges of K n are ordered from 1 . . . � , is there always a long 2 monotone walk? Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing walk of length n − 1. Proof. 1 3 2
Monotone walks: lower bound Question (Chv´ atal-Komlos 1971) � n If edges of K n are ordered from 1 . . . � , is there always a long 2 monotone walk? Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing walk of length n − 1. Proof. 1 3 2
Monotone walks: upper bound Theorem (Graham-Kleitman 1973) There is an edge-ordering of K n in which the longest monotone walk has length n − 1, for all n �∈ { 3 , 5 } .
Monotone walks: upper bound Theorem (Graham-Kleitman 1973) There is an edge-ordering of K n in which the longest monotone walk has length n − 1, for all n �∈ { 3 , 5 } . Proof (for even n ). Edges of K n can be partitioned into perfect matchings.
Monotone walks: upper bound Theorem (Graham-Kleitman 1973) There is an edge-ordering of K n in which the longest monotone walk has length n − 1, for all n �∈ { 3 , 5 } . Proof (for even n ). Edges of K n can be partitioned into perfect matchings. 1 3 4 5 6 2 Assign a batch of consecutive labels to each matching.
Self-avoiding walks Definition A path in a graph is a self-avoiding walk , which never visits the same vertex twice.
Self-avoiding walks Definition A path in a graph is a self-avoiding walk , which never visits the same vertex twice. Self-avoiding walks are more complicated Easy poly-time algorithm to find longest increasing walk.
Self-avoiding walks Definition A path in a graph is a self-avoiding walk , which never visits the same vertex twice. Self-avoiding walks are more complicated Easy poly-time algorithm to find longest increasing walk. In probability: self-avoiding random walk proven sub-ballistic only in 2012 by Duminil-Copin and Hammond.
Monotone paths Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing path of length √ n − 1.
Monotone paths Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing path of length √ n − 1. Proof. Employ walkers again. When edge called, if a walker would revisit a vertex, neither walker moves.
Monotone paths Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing path of length √ n − 1. Proof. Employ walkers again. When edge called, if a walker would revisit a vertex, neither walker moves. Suppose all walkers take ≤ k steps. At most kn 2 edges are walked.
Monotone paths Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing path of length √ n − 1. Proof. Employ walkers again. When edge called, if a walker would revisit a vertex, neither walker moves. Suppose all walkers take ≤ k steps. At most kn 2 edges are walked. � − k = � edges. � k + 1 � k Each walker refuses at most 2 2
Monotone paths Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing path of length √ n − 1. Proof. Employ walkers again. When edge called, if a walker would revisit a vertex, neither walker moves. Suppose all walkers take ≤ k steps. At most kn 2 edges are walked. � − k = � edges. � k + 1 � k Each walker refuses at most 2 2 � � � � n = k 2 n n = walked + refused ≤ kn k 2 + 2 2 2
Monotone paths Theorem (Graham-Kleitman 1973) Every edge-ordering of K n has an increasing path of length √ n − 1. Proof. Employ walkers again. When edge called, if a walker would revisit a vertex, neither walker moves. Suppose all walkers take ≤ k steps. At most kn 2 edges are walked. � − k = � edges. � k + 1 � k Each walker refuses at most 2 2 � � � � n = k 2 n n = walked + refused ≤ kn k 2 + 2 2 2 Theorem (Calderbank-Chung-Sturtevant 1984) There is an edge-ordering of K n in which the longest increasing path has length ( 1 2 − o ( 1 )) n .
Random ordering Model � edges. � n Sample uniformly random ordering of 2
Random ordering Model � edges. � n Sample uniformly random ordering of 2 Equiv: assign independent Unif [ 0 , 1 ] random real to each edge.
Random ordering Model � edges. � n Sample uniformly random ordering of 2 Equiv: assign independent Unif [ 0 , 1 ] random real to each edge. Observation A random edge-ordering has an increasing path of length at least ( 1 − 1 e ) n a.a.s.
Random ordering Model � edges. � n Sample uniformly random ordering of 2 Equiv: assign independent Unif [ 0 , 1 ] random real to each edge. Observation A random edge-ordering has an increasing path of length at least ( 1 − 1 e ) n a.a.s. Proof sketch. Start at arbitrary vertex, expose labels of incident edges. Smallest incident label is min of n − 1 Uniforms, so expectation is 1 n .
Random ordering Model � edges. � n Sample uniformly random ordering of 2 Equiv: assign independent Unif [ 0 , 1 ] random real to each edge. Observation A random edge-ordering has an increasing path of length at least ( 1 − 1 e ) n a.a.s. Proof sketch. Start at arbitrary vertex, expose labels of incident edges. Smallest incident label is min of n − 1 Uniforms, so expectation is 1 n . Take that edge, then expose labels of edges to n − 2 remaining vertices. 1 Smallest increment is min of n − 2 Unifs, so expectation n − 1 .
Random ordering Model � edges. � n Sample uniformly random ordering of 2 Equiv: assign independent Unif [ 0 , 1 ] random real to each edge. Observation A random edge-ordering has an increasing path of length at least ( 1 − 1 e ) n a.a.s. Proof sketch. Start at arbitrary vertex, expose labels of incident edges. Smallest incident label is min of n − 1 Uniforms, so expectation is 1 n . Take that edge, then expose labels of edges to n − 2 remaining vertices. 1 Smallest increment is min of n − 2 Unifs, so expectation n − 1 . Sum 1 n − 1 + · · · + 1 1 cn = 1 when log 1 n + c = 1. �
Random ordering: upper bound Trivial bound A.a.s., a random edge-ordering does not have a Hamiltonian increasing path.
Random ordering: upper bound Trivial bound A.a.s., a random edge-ordering does not have a Hamiltonian increasing path. Proof. (first moment method) For a given Hamiltonian path, it is increasing with probability 1 ( n − 1 )! .
Random ordering: upper bound Trivial bound A.a.s., a random edge-ordering does not have a Hamiltonian increasing path. Proof. (first moment method) For a given Hamiltonian path, it is increasing with probability 1 ( n − 1 )! . Number of Hamiltonian paths is n ! .
Random ordering: upper bound Trivial bound A.a.s., a random edge-ordering does not have a Hamiltonian increasing path. Proof. (first moment method) For a given Hamiltonian path, it is increasing with probability 1 ( n − 1 )! . Number of Hamiltonian paths is n ! . Expected number of increasing Hamiltonian paths is n . . .
In Erd˝ os-R´ enyi First moment insufficient G n , p has n Hamiltonian paths on expectation when n ! p n − 1 ∼ n , i.e., when p ∼ e n .
In Erd˝ os-R´ enyi First moment insufficient G n , p has n Hamiltonian paths on expectation when n ! p n − 1 ∼ n , i.e., when p ∼ e n . Theorem (Bollob´ as) A.a.s., random graph process gets Hamiltonian cycle at moment that all vertices have degree ≥ 2, which is at p ∼ log n + log log n + ω . n
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