Yaocao example updated with notes

CP/optimization/100potions.dzn

@@ -1,10 +1,15 @@
-n = 100; %Water Number
-w = 8; %Window Size
-p = 140; %Leaves required in each Window
-capacity= 380; %Total Energy we have for using holy Water
-m= 4; %Mode numbers
+n = 100;        % number of potions, i.e. the number of segments we will grow on the tree
+m = 4;          % maximum number of segments emerging from a node (depends on the potion applied, see "nutrient")
+w = 8;          % window size, a segment count (evidently w <= n)
+p = 140;        % min number of leaves required in each window
+capacity = 380; % total capacity of nutrients of a path through the tree
 
-nutrient = 
+% For each potion (row) applied to a tree node, the nutrients
+% required by each segment growing from that tree node (col).
+% 381 is "beyond the capacity of nutrients" so essentially a
+% stand-in for "infinity", that is, "that segment does not exist".
+
+nutrient =
 [|6,5,381,381
 |4,1,5,381
 |5,6,3,381
@@ -107,7 +112,11 @@ nutrient =
 |4,381,381,381
 |];
 
-leave = 
+% For each potion (row) applied to a tree node, the number of
+% leaves generated by each segment growing from that tree node (col).
+% For unavailable segments, the number of leaves is 0.
+
+leave =
 [|68,36,0,0
 |18,48,32,0
 |83,90,18,0
@@ -208,4 +217,4 @@ leave =
 |93,35,0,0
 |6,3,51,0
 |19,0,0,0
-|];
+|];

CP/optimization/4potions.dzn

@@ -1,19 +1,28 @@
-n = 4;
-m = 4;
-w = 2;
-p = 8;
-capacity= 19;
+n = 4;         % number of potions, i.e. the number of segments we will grow on the tree
+m = 4;         % maximum number of segments emerging from a node (depends on the potion applied, see "nutrient")
+w = 2;         % window size, a segment count (evidently w <= n)
+p = 8;         % min number of leaves required in each window
+capacity = 19; % total capacity of nutrients of a path through the tree
 
-nutrient = 
+% For each potion (row) applied to a tree node, the nutrients
+% required by each segment growing from that tree node (col).
+% 20 is "beyond the capacity of nutrients" so essentially a
+% stand-in for "infinity", that is, "that segment does not exist".
+
+nutrient =
 [|4,2,3,7
 |3,8,20,20
 |5,4,6,20
 |3,6,2,20
 |];
 
-leave = 
+% For each potion (row) applied to a tree node, the number of
+% leaves generated by each segment growing from that tree node (col).
+% For unavailable segments, the number of leaves is 0.
+
+leave =
 [|7,6,4,8
 |5,8,0,0
 |5,4,7,0
 |6,9,3,0
-|];
+|];

CP/optimization/yaocao.mzn

@@ -1,31 +1,74 @@
-int: n; % number of potions
+% Solving the "Yao Cao" problem
+% https://www.coursera.org/learn/solving-algorithms-discrete-optimization/lecture/t2J76/3-2-1-optimization-in-cp
+
+int: n;                      % number of potions, i.e. the number of segments we will grow on the tree
 set of int: POTIONS = 1..n;
-int: w; % window size
-int: p; % num of leaves in each window
-int: capacity; % capacity of nutrients
-int: m; % max number of grown segments
+int: w;                      % window size, a segment count (evidently w <= n)
+int: p;                      % min number of leaves required in each window
+int: capacity;               % total capacity of nutrients of a path through the tree
+int: m;                      % maximum number of segments emerging from a node (depends on the potion applied, see "nutrient")
 set of int: SEGMENTS = 1..m;
 
-array[POTIONS,SEGMENTS] of int: nutrient;   % nutrient required
-array[POTIONS,SEGMENTS] of int: leave; % num of leaves to grow
+constraint assert(n>=w,"Window Size larger than number of potions (aka., number of segments)");
+
+% For each potion (row) applied to a tree node, the nutrients
+% required by each segment growing from that tree node (col).
+% If the nutrient value is "beyond the capacity of nutrients", the
+% meaning is "that segment does not exist" as we immediately
+% would violate the nutrient limit if that segment were chosen-.
+
+array[POTIONS,SEGMENTS] of int: nutrient;
+
+% For each potion (row) applied to a tree node, the number of
+% leaves generated by each segment growing from that tree node (col).
+% For unavailable segments, the number of leaves is 0.
+
+array[POTIONS,SEGMENTS] of int: leave;
+
+% What we are looking for: for each potion applied to a
+% tree node, we select the next segment to traverse. We do not
+% have a choice in potion order, we have to use the potions
+% in order of appearance.
+
+array[POTIONS] of var SEGMENTS: choice;
 
-array[POTIONS] of var SEGMENTS: choice; %Choice of segment for each potion
-array[POTIONS] of var int: nutrient_list = 
+% Derived: the number of nutrients required for each segment,
+% which his identified by the potion applied to its parent tree node.
+
+array[POTIONS] of var int: nutrient_list =
     [nutrient[i,choice[i]] | i in POTIONS];
-array[POTIONS] of var int: leave_list = 
+
+% Derived: the number of leaves generated by each segment,
+% which his identified by the potion applied to its parent tree node.
+
+array[POTIONS] of var int: leave_list =
     [leave[i,choice[i]] | i in POTIONS];
 
 var int: total_nutrient = sum(nutrient_list);
 var int: total_leaves = sum(leave_list);
 
-constraint assert(n>=w,"Window Size larger than # of steps");
+% Can't exceed the total nutrient capacity
 
-%Capacity
 constraint total_nutrient <= capacity;
-%In each window, leave numbers should be larger than p
+
+% In each window, number of leaves must reach the minimum
+
 constraint forall(tail in w..n)
     (sum(i in tail-w+1..tail)(leave_list[i]) >= p);
 
+% Solving
+% https://docs.minizinc.dev/en/stable/lib-stdlib-annotations.html#seq-search
+% https://docs.minizinc.dev/en/stable/lib-stdlib-annotations.html#int-search
+% On 100potions.dzn with "12th Gen Intel(R) Core(TM) i3-12100" + "Gecode 6.3.0":
+% To find total_leaves = 6443; total_nutrient = 380:
+% ssearch1         ~  9 mins
+% ssearch2+search1 ~ 27 s
+% ssearch2+search2 ~ 27 s
+% ssearch2+search3 ~ 15 s
+% ssearch2+search4 ~ 11 s
+% ssearch2+search5 ~  9 s
+% ssearch2+search6 ~ 19 s
+
 solve :: ssearch2 maximize total_leaves;
 
 ann: search1 = int_search(leave_list, input_order, indomain_min, complete);
@@ -37,8 +80,14 @@ ann: search6 = int_search(leave_list, smallest, indomain_max, complete);
 
 ann: ssearch1 = seq_search([search1,
     int_search([total_leaves], input_order, indomain_max, complete) ]);
+
 ann: ssearch2 = seq_search([
     int_search([total_leaves], input_order, indomain_max, complete),
     search1 ]);
 
-output ["choice = ",show(choice),";\n","total_leaves = ",show(total_leaves),";\n","total_nutrient = ", show(total_nutrient)];
+output ["choice = ",show(choice),";\n","total_leaves = ",show(total_leaves),";\n","total_nutrient = ", show(total_nutrient)];
+output ["\n"];
+output ["Apply potion \(i)"++
+        ", then select segment \(fix(choice[i]))"++
+        " which needs \(nutrient[i,fix(choice[i])]) nutrients"++
+        " and gives \(leave[i,fix(choice[i])]) leaves\n" | i in POTIONS];