0.06/0.22 WORST_CASE(?,O(n^2)) 0.06/0.22 0.06/0.22 Preprocessing Cost Relations 0.06/0.22 ===================================== 0.06/0.22 0.06/0.22 #### Computed strongly connected components 0.06/0.22 0. recursive : [eval_realheapsort_step1_2/3,eval_realheapsort_step1_3/4,eval_realheapsort_step1_4/5,eval_realheapsort_step1_5/5,eval_realheapsort_step1_6/5,eval_realheapsort_step1_bb2_in/3,eval_realheapsort_step1_bb3_in/3,eval_realheapsort_step1_bb4_in/5] 0.06/0.22 1. recursive : [eval_realheapsort_step1__critedge_in/4,eval_realheapsort_step1_bb1_in/3,eval_realheapsort_step1_bb2_in_loop_cont/5] 0.06/0.22 2. non_recursive : [eval_realheapsort_step1_stop/1] 0.06/0.22 3. non_recursive : [eval_realheapsort_step1_bb5_in/1] 0.06/0.22 4. non_recursive : [eval_realheapsort_step1_bb1_in_loop_cont/2] 0.06/0.22 5. non_recursive : [eval_realheapsort_step1_bb0_in/2] 0.06/0.22 6. non_recursive : [eval_realheapsort_step1_start/2] 0.06/0.22 0.06/0.22 #### Obtained direct recursion through partial evaluation 0.06/0.22 0. SCC is partially evaluated into eval_realheapsort_step1_bb2_in/3 0.06/0.22 1. SCC is partially evaluated into eval_realheapsort_step1_bb1_in/3 0.06/0.22 2. SCC is completely evaluated into other SCCs 0.06/0.22 3. SCC is completely evaluated into other SCCs 0.06/0.22 4. SCC is completely evaluated into other SCCs 0.06/0.22 5. SCC is partially evaluated into eval_realheapsort_step1_bb0_in/2 0.06/0.22 6. SCC is partially evaluated into eval_realheapsort_step1_start/2 0.06/0.22 0.06/0.22 Control-Flow Refinement of Cost Relations 0.06/0.22 ===================================== 0.06/0.22 0.06/0.22 ### Specialization of cost equations eval_realheapsort_step1_bb2_in/3 0.06/0.22 * CE 7 is refined into CE [9] 0.06/0.22 * CE 8 is refined into CE [10] 0.06/0.22 * CE 6 is refined into CE [11] 0.06/0.22 0.06/0.22 0.06/0.22 ### Cost equations --> "Loop" of eval_realheapsort_step1_bb2_in/3 0.06/0.22 * CEs [11] --> Loop 9 0.06/0.22 * CEs [9] --> Loop 10 0.06/0.22 * CEs [10] --> Loop 11 0.06/0.22 0.06/0.22 ### Ranking functions of CR eval_realheapsort_step1_bb2_in(V_j_0,B,C) 0.06/0.22 * RF of phase [9]: [V_j_0] 0.06/0.22 0.06/0.22 #### Partial ranking functions of CR eval_realheapsort_step1_bb2_in(V_j_0,B,C) 0.06/0.22 * Partial RF of phase [9]: 0.06/0.22 - RF of loop [9:1]: 0.06/0.22 V_j_0 0.06/0.22 0.06/0.22 0.06/0.22 ### Specialization of cost equations eval_realheapsort_step1_bb1_in/3 0.06/0.22 * CE 5 is refined into CE [12] 0.06/0.22 * CE 4 is refined into CE [13,14,15] 0.06/0.22 0.06/0.22 0.06/0.22 ### Cost equations --> "Loop" of eval_realheapsort_step1_bb1_in/3 0.06/0.22 * CEs [13,14] --> Loop 12 0.06/0.22 * CEs [15] --> Loop 13 0.06/0.22 * CEs [12] --> Loop 14 0.06/0.22 0.06/0.22 ### Ranking functions of CR eval_realheapsort_step1_bb1_in(V_N,V_k_0,B) 0.06/0.22 * RF of phase [12,13]: [V_N-V_k_0] 0.06/0.22 0.06/0.22 #### Partial ranking functions of CR eval_realheapsort_step1_bb1_in(V_N,V_k_0,B) 0.06/0.22 * Partial RF of phase [12,13]: 0.06/0.22 - RF of loop [12:1,13:1]: 0.06/0.22 V_N-V_k_0 0.06/0.22 0.06/0.22 0.06/0.22 ### Specialization of cost equations eval_realheapsort_step1_bb0_in/2 0.06/0.22 * CE 3 is refined into CE [16] 0.06/0.22 * CE 2 is refined into CE [17] 0.06/0.22 0.06/0.22 0.06/0.22 ### Cost equations --> "Loop" of eval_realheapsort_step1_bb0_in/2 0.06/0.22 * CEs [16] --> Loop 15 0.06/0.22 * CEs [17] --> Loop 16 0.06/0.22 0.06/0.22 ### Ranking functions of CR eval_realheapsort_step1_bb0_in(V_N,B) 0.06/0.22 0.06/0.22 #### Partial ranking functions of CR eval_realheapsort_step1_bb0_in(V_N,B) 0.06/0.22 0.06/0.22 0.06/0.22 ### Specialization of cost equations eval_realheapsort_step1_start/2 0.06/0.22 * CE 1 is refined into CE [18,19] 0.06/0.22 0.06/0.22 0.06/0.22 ### Cost equations --> "Loop" of eval_realheapsort_step1_start/2 0.06/0.22 * CEs [19] --> Loop 17 0.06/0.22 * CEs [18] --> Loop 18 0.06/0.22 0.06/0.22 ### Ranking functions of CR eval_realheapsort_step1_start(V_N,B) 0.06/0.22 0.06/0.22 #### Partial ranking functions of CR eval_realheapsort_step1_start(V_N,B) 0.06/0.22 0.06/0.22 0.06/0.22 Computing Bounds 0.06/0.22 ===================================== 0.06/0.22 0.06/0.22 #### Cost of chains of eval_realheapsort_step1_bb2_in(V_j_0,B,C): 0.06/0.22 * Chain [[9],11]: 1*it(9)+0 0.06/0.22 Such that:it(9) =< V_j_0 0.06/0.22 0.06/0.22 with precondition: [B=2,0>=C,V_j_0>=1,2*C+1>=0] 0.06/0.22 0.06/0.22 * Chain [[9],10]: 1*it(9)+0 0.06/0.22 Such that:it(9) =< V_j_0-C 0.06/0.22 0.06/0.22 with precondition: [B=2,C>=1,V_j_0>=2*C+1] 0.06/0.22 0.06/0.22 * Chain [10]: 0 0.06/0.22 with precondition: [B=2,V_j_0=C,V_j_0>=1] 0.06/0.22 0.06/0.22 0.06/0.22 #### Cost of chains of eval_realheapsort_step1_bb1_in(V_N,V_k_0,B): 0.06/0.22 * Chain [[12,13],14]: 2*it(12)+1*s(5)+1*s(6)+0 0.06/0.22 Such that:aux(1) =< V_N 0.06/0.22 aux(5) =< V_N-V_k_0 0.06/0.22 it(12) =< aux(5) 0.06/0.22 aux(2) =< aux(1) 0.06/0.22 s(5) =< it(12)*aux(1) 0.06/0.22 s(6) =< it(12)*aux(2) 0.06/0.22 0.06/0.22 with precondition: [B=3,V_N>=3,V_k_0>=1,V_N>=V_k_0+1] 0.06/0.22 0.06/0.22 0.06/0.22 #### Cost of chains of eval_realheapsort_step1_bb0_in(V_N,B): 0.06/0.22 * Chain [16]: 0 0.06/0.22 with precondition: [2>=V_N] 0.06/0.22 0.06/0.22 * Chain [15]: 2*s(9)+1*s(11)+1*s(12)+0 0.06/0.22 Such that:aux(6) =< V_N 0.06/0.22 s(9) =< aux(6) 0.06/0.22 s(10) =< aux(6) 0.06/0.22 s(11) =< s(9)*aux(6) 0.06/0.22 s(12) =< s(9)*s(10) 0.06/0.22 0.06/0.22 with precondition: [V_N>=3] 0.06/0.22 0.06/0.22 0.06/0.22 #### Cost of chains of eval_realheapsort_step1_start(V_N,B): 0.06/0.22 * Chain [18]: 0 0.06/0.22 with precondition: [2>=V_N] 0.06/0.22 0.06/0.22 * Chain [17]: 2*s(14)+1*s(16)+1*s(17)+0 0.06/0.22 Such that:s(13) =< V_N 0.06/0.22 s(14) =< s(13) 0.06/0.22 s(15) =< s(13) 0.06/0.22 s(16) =< s(14)*s(13) 0.06/0.22 s(17) =< s(14)*s(15) 0.06/0.22 0.06/0.22 with precondition: [V_N>=3] 0.06/0.22 0.06/0.22 0.06/0.22 Closed-form bounds of eval_realheapsort_step1_start(V_N,B): 0.06/0.22 ------------------------------------- 0.06/0.22 * Chain [18] with precondition: [2>=V_N] 0.06/0.22 - Upper bound: 0 0.06/0.22 - Complexity: constant 0.06/0.22 * Chain [17] with precondition: [V_N>=3] 0.06/0.22 - Upper bound: 2*V_N*V_N+2*V_N 0.06/0.22 - Complexity: n^2 0.06/0.22 0.06/0.22 ### Maximum cost of eval_realheapsort_step1_start(V_N,B): nat(V_N)*2*nat(V_N)+nat(V_N)*2 0.06/0.22 Asymptotic class: n^2 0.06/0.22 * Total analysis performed in 137 ms. 0.06/0.22 0.06/0.32 EOF