Breaking the Linear-Memory Barrier in Massively Parallel Computing - PowerPoint PPT Presentation

𝑵 machines Local Memory in MPC 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes. often trivial usual assumption for most problems, only direct simulation of for many problems, often unrealistic LOCAL/PRAM algorithms ▪ ෩ admits O(1)-round algorithms O(𝑜) prohibitively large known based on very simple ▪ sparse graphs trivial sampling approach Algorithms have been stuck at this linear-memory barrier! Lattanzi et al. [SPAA’11]

𝑵 machines Local Memory in MPC 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes. often trivial usual assumption for most problems, only direct simulation of for many problems, often unrealistic LOCAL/PRAM algorithms ▪ ෩ admits O(1)-round algorithms O(𝑜) prohibitively large known based on very simple ▪ sparse graphs trivial sampling approach Algorithms have been stuck at this linear-memory barrier! Lattanzi et al. [SPAA’11] Fundamentally?

Breaking the Linear-Memory Barrier:

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory Ghaffari, Kuhn, Uitto [FOCS’19] 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines Conditional Lower Bound Ω log log 𝑜 rounds poly lo log lo log 𝒐 rounds

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: best we can hope for enhance LOCAL algorithms with global communication GKU [FOCS’19] ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Breaking the Linear-Memory Barrier: Efficient MPC Graph Algorithms with Strongly Sublinear Memory 𝑻 = 𝑷 𝒐 𝜺 local memory 𝑵 = 𝑷 𝒏/𝒐 𝜺 machines poly lo log lo log 𝒐 rounds Imposed Locality: machines see only subset of nodes, regardless of sparsity of graph Our Approach to Cope with Locality: enhance LOCAL algorithms with global communication ▪ exponentially faster than LOCAL algorithms due to shortcuts ▪ polynomially less memory than most MPC algorithms

Problem: Maximal Independent Set (MIS)

Maximal Independent Set (MIS)

Maximal Independent Set (MIS)

Maximal Independent Set (MIS)

Maximal Independent Set (MIS) Independent Set: set of non-adjacent nodes Maximal: no node can be added without violating independence

Maximal Independent Set (MIS) Independent Set: set of non-adjacent nodes Maximal: no node can be added without violating independence

𝑵 machines MIS: State of the Art 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes.

𝑵 machines MIS: State of the Art 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes. Lattanzi et al. [SPAA’11] 𝑃 1

𝑵 machines MIS: State of the Art 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes. Ghaffari et al. [PODC’18] Lattanzi et al. [SPAA’11] 𝑃(log log 𝑜) 𝑃 1

𝑵 machines MIS: State of the Art 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes. Luby’s Algorithm Ghaffari et al. [PODC’18] Lattanzi et al. [SPAA’11] 𝑃(log 𝑜) 𝑃(log log 𝑜) 𝑃 1

𝑵 machines MIS: State of the Art 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes. Luby’s Algorithm Ghaffari et al. [PODC’18] Lattanzi et al. [SPAA’11] 𝑃(log 𝑜) 𝑃(log log 𝑜) 𝑃 1 Ghaffari and Uitto [SODA’19] ෨ 𝑃 log 𝑜

𝑵 machines MIS: State of the Art on Trees 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes. Luby’s Algorithm Ghaffari et al. [PODC’18] Lattanzi et al. [SPAA’11] 𝑃(log 𝑜) 𝑃(log log 𝑜) 𝑃 1 Ghaffari and Uitto [SODA’19] ෨ 𝑃 log 𝑜

𝑵 machines MIS: State of the Art on Trees 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes. Luby’s Algorithm Trivial solution Ghaffari et al. [PODC’18] Lattanzi et al. [SPAA’11] Trivial solution 𝑃(log 𝑜) 𝑃(log log 𝑜) 𝑃 1 𝑃 1 𝑃 1 Ghaffari and Uitto [SODA’19] ෨ 𝑃 log 𝑜

𝑵 machines MIS: State of the Art on Trees 𝑻 memory per machine 𝑵 ⋅ 𝑻 = ෩ 𝑷 (𝒏 + 𝒐) S ෩ ෩ ෩ O 𝑜 𝜀 Ω 𝑜 1+𝜀 Θ(𝑜) Strongly Sublinear Memory: Linear Memory: Superlinear Memory: 𝑃 𝑜 𝜀 , 0 ≤ 𝜀 < 1 𝑃 𝑜 1+𝜀 , 0 < 𝜀 ≤ 1 𝑇 = ෨ 𝑇 = ෨ 𝑇 = ෨ 𝑃 𝑜 Machines see all nodes. No machine sees all nodes. Machines see all nodes. Our Result Luby’s Algorithm Trivial solution Ghaffari et al. [PODC’18] Lattanzi et al. [SPAA’11] Trivial solution 𝑃 log 3 log 𝑜 𝑃(log 𝑜) 𝑃 1 𝑃(log log 𝑜) 𝑃 1 𝑃 1 Ghaffari and Uitto [SODA’19] ෨ 𝑃 log 𝑜

Our Result 𝑷(𝐦𝐩𝐡 𝟒 𝐦𝐩𝐡 𝒐) -round MPC algorithm 𝑷 𝒐 𝜺 memory that w.h.p. computes MIS on trees. with 𝐓 = ෩

Our Result ෩ 𝑷 𝐦𝐩𝐡 𝒐 rounds 𝑷 𝒐 𝜺 memory 𝐓 = ෩ Ghaffari and Uitto [SODA’19] 𝑷(𝐦𝐩𝐡 𝟒 𝐦𝐩𝐡 𝒐) -round MPC algorithm 𝑷 𝒐 𝜺 memory that w.h.p. computes MIS on trees. with 𝐓 = ෩

Our Result ෩ 𝑷 𝐦𝐩𝐡 𝐦𝐩𝐡 𝒐 rounds 𝑷 𝐦𝐩𝐡 𝒐 rounds 𝐓 = ෩ 𝑷 𝒐 𝜺 memory 𝐓 = ෩ 𝑷 𝒐 memory Ghaffari et al. [PODC’18] Ghaffari and Uitto [SODA’19] 𝑷(𝐦𝐩𝐡 𝟒 𝐦𝐩𝐡 𝒐) -round MPC algorithm 𝑷 𝒐 𝜺 memory that w.h.p. computes MIS on trees. with 𝐓 = ෩

Our Result ෩ 𝑷 𝐦𝐩𝐡 𝐦𝐩𝐡 𝒐 rounds 𝑷 𝐦𝐩𝐡 𝒐 rounds 𝐓 = ෩ 𝑷 𝒐 𝜺 memory 𝐓 = ෩ 𝑷 𝒐 memory Ghaffari et al. [PODC’18] Ghaffari and Uitto [SODA’19] 𝑷(𝐦𝐩𝐡 𝟒 𝐦𝐩𝐡 𝒐) -round MPC algorithm 𝑷 𝒐 𝜺 memory that w.h.p. computes MIS on trees. with 𝐓 = ෩ Conditional 𝛁(𝐦𝐩𝐡 𝐦𝐩𝐡 𝒐) -round lower bound for 𝐓 = ෩ 𝑷 𝒐 𝜺 Ghaffari, Kuhn, and Uitto [FOCS’19]

Algorithm

Algorithm Outline

Algorithm Outline

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction ii) LOCAL Shattering Ghaffari [SODA’16] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction main LOCAL technique ii) LOCAL Shattering Ghaffari [SODA’16] Beck [RSA’91] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction ii) LOCAL Shattering Ghaffari [SODA’16] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction ii) LOCAL Shattering Ghaffari [SODA’16] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Algorithm Outline 1) Shattering break graph into small components i) Degree Reduction ii) LOCAL Shattering Ghaffari [SODA’16] 2) Post-Shattering solve problem on remaining components i) Gathering of Components ii) Local Computation

Polynomial Degree Reduction: Subsample-and-Conquer

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring

Polynomial Degree Reduction: Subsample-and-Conquer Subsample subsample nodes independently Conquer compute random MIS in subsampled graph ▪ gather connected components ▪ locally compute random 2-coloring ▪ add a color class to MIS