Self-Adjusting Grid Networks to Minimize Expected Path Length Chen Avin, Michael Borokhovich, Bernhard Haeupler, Zvi Lotker Communication Systems Engineering, BGU, Israel Computer Science and Artificial Intelligence Laboratory, MIT, USA SIROCCO 2013 Motivation - Data Centers Energy cost ($50B in US alone 2008, doubles every 5 years!) Routing consumes about 20-30% Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 1 / 21 Motivation - Data Centers Energy cost ($50B in US alone 2008, doubles every 5 years!) Routing consumes about 20-30% Need to adjust the network, i.e., reduce the expected route length Fixed infrastructure... Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 1 / 21 Motivation - Data Centers Energy cost ($50B in US alone 2008, doubles every 5 years!) Routing consumes about 20-30% Need to adjust the network, i.e., reduce the expected route length Fixed infrastructure... Move processes (e.g., VM) between machines Virtualization and SDN (software defined networks), e.g., OpenFlow enable VM migration Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 1 / 21 Simple Example P4 P1 P6 P2 P5 Michael Borokhovich P3 P1 P2 P4 P3 P5 P6 Self-Adjusting Grid Networks to Minimize Expected Path Length 1 2 1 5 3 10 2 / 21 Simple Example P4 P1 P6 P2 P5 P3 E [route length] = 12 4 + 15 6 + Michael Borokhovich 3 10 4 P1 P2 P4 P3 P5 P6 1 2 1 5 3 10 = 4.4 Self-Adjusting Grid Networks to Minimize Expected Path Length 2 / 21 Simple Example P4 P3 P1 P2 P6 P5 Michael Borokhovich P1 P2 P4 P3 P5 P6 E [route length] = 12 4 + 15 6 + 3 10 4 = 4.4 E [route length] = 12 1 + 15 1 + 3 10 1 =1 Self-Adjusting Grid Networks to Minimize Expected Path Length 1 2 1 5 3 10 2 / 21 Model and Problem Definition Host Graph: H(V , E): Physical Infrastructure Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 3 / 21 Model and Problem Definition Host Graph: H(V , E): Physical Infrastructure Routing Requests: σ = (σ1 , σ2 , . . . , σm ) Michael Borokhovich σt = (u, v ) Self-Adjusting Grid Networks to Minimize Expected Path Length 3 / 21 Model and Problem Definition Host Graph: H(V , E): Physical Infrastructure Routing Requests: σ = (σ1 , σ2 , . . . , σm ) σt = (u, v ) We assume the requests are i.i.d. from a given distribution Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 3 / 21 Model and Problem Definition Host Graph: H(V , E): Physical Infrastructure 1 1 2 2 Requests Distribution 4 5 1/ 2 0 3 4 3 1/ 8 0 1/ 9 5 Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 4 / 21 Model and Problem Definition Host Graph: H(V , E): Physical Infrastructure 1 1 2 Requests Distribution 3 2 1 1/ 8 1 2 3 4 2 4 1/ 9 5 5 4 0 1/8 5 0 0 3 4 5 3 1/ 2 0 1/2 G(P, W) 1/8 1/9 Guest Weighted Graph: G(P, W ) |V | = |P| = n Michael Borokhovich 1/2 Self-Adjusting Grid Networks to Minimize Expected Path Length p(u, v) 4 / 21 Expected Path Length A placement (arrangement): ϕ:P→V Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 5 / 21 Expected Path Length A placement (arrangement): ϕ:P→V For H, G, ϕ: EPL(ϕ) = X Pr(u, v )dH (ϕ(u), ϕ(v )) u,v ∈P Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 5 / 21 Expected Path Length A placement (arrangement): ϕ:P→V For H, G, ϕ: EPL(ϕ) = X Pr(u, v )dH (ϕ(u), ϕ(v )) u,v ∈P Minimum Expected Path Length Problem MEPL = min EPL(ϕ) ϕ Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 5 / 21 Example G(P, W ) ϕ:P→V Find the best way to put processes on the graph to minimize expected path length H(V , E) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 6 / 21 Related Work 1 2 3 4 5 1 2 VLSI layout 1/8 0 0 1/2 3 4 Minimum Linear Arrangement (MLA) 5 Known to be hard (NP-Complete) 11 22 33 44 5 66 77 88 G(P, W) 99 10 10 1/8 1/9 1/2 MLA = min ϕ Michael Borokhovich X w(u, v )|ϕ(u) − ϕ(v )| u,v ,∈P Self-Adjusting Grid Networks to Minimize Expected Path Length 7 / 21 Hardness of MEPL Host Graph – Grid Guest Graph – Symmetric Product Distribution Activity level: p(u) Probability of request: p(u, v ) = p(u) · p(v ) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 8 / 21 Hardness of MEPL Host Graph – Grid Guest Graph – Symmetric Product Distribution Activity level: p(u) Probability of request: p(u, v ) = p(u) · p(v ) Lemma If G is a symmetric product distribution, MEPL is still hard. Lemma If H is a 2-dimensional grid, MEPL is still hard. Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 8 / 21 Is there a CLIQUE of size k in H ? Lemma If G is a symmetric product distribution, MEPL is still hard. 1 k2 ? ←− k nodes Clique (n − k) disjoint nodes p(u) = 1/k p(u) = 0 Host H Guest G arbitrary graph symmetric product distribution p(u, v ) = p(u) · p(v ) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 9 / 21 Is there a CLIQUE of size k in H ? Lemma If G is a symmetric product distribution, MEPL is still hard. 1 k2 ? ←− k nodes Clique (n − k) disjoint nodes p(u) = 1/k p(u) = 0 Host H Guest G arbitrary graph symmetric product distribution p(u, v ) = p(u) · p(v ) H has a clique of size k if and only if MEPL = Michael Borokhovich k (k −1) k2 Self-Adjusting Grid Networks to Minimize Expected Path Length =1− 1 k 9 / 21 Can we embed Tree into Grid? Lemma If H is a 2-dimensional grid, MEPL is still hard. ? ←− 1 k −1 Guest G Guest G tree (k nodes) Can we embed G to H? This is hard [Bhatt et al., 1987] Host H Host H grid (k 2 nodes) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 10 / 21 Can we embed Tree into Grid? Lemma If H is a 2-dimensional grid, MEPL is still hard. ? ←− 1 k −1 Guest G Guest G tree (k nodes) Can we embed G to H? This is hard [Bhatt et al., 1987] Host H Host H grid (k 2 nodes) G can be embedded into H if and only if MEPL = Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length k −1 k −1 =1 10 / 21 Main Result Theorem For d-dimensional grid (H) and a symmetric product distribution (G) there is a simple distributed algorithm with a local switching policy between processes and their neighbors that achieves a constant approximation to MEPL Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 11 / 21 Expected Distance to Center Expected center: c ∗ (ϕ) = arg min x X p(u)d(ϕ(u), ϕ(x)) u ϕ(u) c∗ ϕ(v ) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 12 / 21 Expected Distance to Center Expected center: c ∗ (ϕ) = arg min Expected distance to center: x X p(u)d(ϕ(u), ϕ(x)) u C(ϕ) = X p(u)d(ϕ(u), c ∗ (ϕ)) u ϕ(u) c∗ ϕ(v ) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 12 / 21 Expected Distance to Center Expected center: c ∗ (ϕ) = arg min Expected distance to center: x X p(u)d(ϕ(u), ϕ(x)) u C(ϕ) = X p(u)d(ϕ(u), c ∗ (ϕ)) u Minimum expected distance: Cmin = min C(ϕ) ϕ ϕ(u) c∗ ϕ(v ) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 12 / 21 Switching Rule – Optimize Expected Distance to the Center ϕ1 (u) switch u and v? c∗ ϕ1 (v ) ϕ2 (v ) −→ c∗ ϕ2 (u) Switch only if: C(ϕ2 ) ≤ C(ϕ1 ) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 13 / 21 Switching Rule – Optimize Expected Distance to the Center ϕ1 (u) switch u and v? ϕ2 (v ) −→ c∗ c∗ ϕ2 (u) ϕ1 (v ) Switch only if: C(ϕ2 ) ≤ C(ϕ1 ) Assumptions: Recall that: C(ϕ) = X p(u)d(ϕ(u), c ∗ (ϕ)) u Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 13 / 21 Switching Rule – Optimize Expected Distance to the Center ϕ1 (u) switch u and v? ϕ2 (v ) −→ c∗ c∗ ϕ2 (u) ϕ1 (v ) Switch only if: C(ϕ2 ) ≤ C(ϕ1 ) Assumptions: Recall that: C(ϕ) = X p(u)d(ϕ(u), c ∗ (ϕ)) u Every node knows current ϕ (locations of all nodes in H) centralized directory Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 13 / 21 Switching Rule – Optimize Expected Distance to the Center ϕ1 (u) switch u and v? ϕ2 (v ) −→ c∗ c∗ ϕ2 (u) ϕ1 (v ) Switch only if: C(ϕ2 ) ≤ C(ϕ1 ) Assumptions: Recall that: C(ϕ) = X p(u)d(ϕ(u), c ∗ (ϕ)) u Every node knows current ϕ (locations of all nodes in H) centralized directory Every node knows activity level p(u) of all nodes observing requests over time Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 13 / 21 Switching Rule – Optimize Expected Distance to the Center Greedy approach Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21 Switching Rule – Optimize Expected Distance to the Center Greedy approach Every switch decreases C(ϕ) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21 Switching Rule – Optimize Expected Distance to the Center Greedy approach Every switch decreases C(ϕ) Will stop at some local minimum placement ϕ b Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21 Switching Rule – Optimize Expected Distance to the Center Greedy approach Every switch decreases C(ϕ) Will stop at some local minimum placement ϕ b How far local minimum C(ϕ) b from global minimum Cmin ? Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21 Switching Rule – Optimize Expected Distance to the Center Greedy approach Every switch decreases C(ϕ) Will stop at some local minimum placement ϕ b How far local minimum C(ϕ) b from global minimum Cmin ? What can we say about EPL(ϕ)? b Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21 Switching Rule – Optimize Expected Distance to the Center Greedy approach Every switch decreases C(ϕ) Will stop at some local minimum placement ϕ b How far local minimum C(ϕ) b from global minimum Cmin ? What can we say about EPL(ϕ)? b We show: C(ϕ) b = O(1) Cmin Michael Borokhovich and EPL(ϕ) b = O(1) MEPL Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21 MEPL and Minimum Expected Distance to Center Lemma ∀ϕ : C(ϕ) ≤ EPL(ϕ) ≤ 2C(ϕ) ϕ(u) c∗ ϕ(v ) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 15 / 21 MEPL and Minimum Expected Distance to Center Lemma ∀ϕ : C(ϕ) ≤ EPL(ϕ) ≤ 2C(ϕ) d(ϕ(u), ϕ(v )) ≤ d(ϕ(u), c ∗ ) + d(c ∗ , ϕ(v )) ϕ(u) c∗ ϕ(v ) Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 15 / 21 Expected Rank Rank of a node r(u) is the position of the node in the ordered list of nodes’ activity levels. Node with the highest activity level has rank 0. E[R] = X p(u)r(u) u Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 16 / 21 Line For any local optimum ϕ: b C(ϕ) b ≤ E[R] p(u) d(ϕ(u), b c ∗ ) ≤ r(u) c∗ Michael Borokhovich ϕ(u) b Self-Adjusting Grid Networks to Minimize Expected Path Length 17 / 21 Line For any local optimum ϕ: b C(ϕ) b ≤ E[R] p(u) d(ϕ(u), b c ∗ ) ≤ r(u) c∗ ϕ(u) b For the global optimum ϕ: e Cmin ≥ 12 E[R] p(u) d(ϕ(u), e c ∗ ) ≥ r(u)/2 c∗ Michael Borokhovich ϕ(u) e Self-Adjusting Grid Networks to Minimize Expected Path Length 17 / 21 2-Dimensional Grid C(ϕ) b ≈ Michael Borokhovich √ n Self-Adjusting Grid Networks to Minimize Expected Path Length 18 / 21 2-Dimensional Grid C(ϕ) b ≈ Michael Borokhovich √ n Cmin ≈ Self-Adjusting Grid Networks to Minimize Expected Path Length √ 4 n 18 / 21 2-Dimensional Grid C(ϕ) b ≈ √ n Cmin ≈ √ 4 n v Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 18 / 21 2-Dimensional Grid C(ϕ) b ≈ √ n Cmin ≈ √ 4 n Allow chess knight moves 6 18 8 13 openclipart.org/people e portable e im m portable e im_Chess_tile_-_Knight_2.sv v v Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 18 / 21 2-Dimensional Grid For any local optimum ϕ b x2 x3 x1 x4 v z1 x5 c∗ x6 z2 z3 x7 z4 z6 C(ϕ) b ≤ z5 √ R] √4 E[ 6 d(ϕ(v b ), c ∗ ) ≤ Michael Borokhovich √4 6 p r(v ) Self-Adjusting Grid Networks to Minimize Expected Path Length 19 / 21 2-Dimensional Grid For any local optimum ϕ b For the global optimum ϕ e x2 x3 x1 x4 v v z1 x5 c∗ c∗ x6 z2 z3 x7 z4 z6 C(ϕ) b ≤ z5 √ R] √4 E[ 6 d(ϕ(v b ), c ∗ ) ≤ Michael Borokhovich √4 6 p r(v ) Cmin ≥ √ √1 E[ 2 d(ϕ(v e ), c ∗ ) ≥ Self-Adjusting Grid Networks to Minimize Expected Path Length q R] r(v ) 2 19 / 21 Simulations – Clustered Requests 900 nodes 50% inactive 8 clusters Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 20 / 21 Simulations – Clustered Requests 900 nodes 50% inactive 8 clusters 900 nodes 16 clusters Animation Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 20 / 21 Summary MEPL is hard for general graphs and requests patterns For grids and symm. product distr. we showed greedy approach that is a constant approximation Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 21 / 21 Summary MEPL is hard for general graphs and requests patterns For grids and symm. product distr. we showed greedy approach that is a constant approximation Future work: Real datacenters infrastructure More requests patterns Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 21 / 21 Summary MEPL is hard for general graphs and requests patterns For grids and symm. product distr. we showed greedy approach that is a constant approximation Future work: Real datacenters infrastructure More requests patterns THANK YOU! Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 21 / 21
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