Approach

SAを使いました。

複数の点の位置を決める問題に+αした問題(最近だと CirclesMix, HHMM2nd, 山型足し算, stick_xor, ...)にもSAがよく使われます。上位の人たちでも点数に差がついて、いい問題であることが多いです。私もSAより貪欲法のほうが良くなったりして、力不足だと思わされるのもこのタイプの問題(CirclesMix等)です。


温度

10 から 1.25 に線形に減らす。


近傍

  • 物を置く
  • 物を移動する
  • 物を取り除く
  • Lantern を強制的に置く(競合している Lantern があったら取り除く)
    • 競合しているせいで、最適化出来てないことがあったので。
  • Incorrect な Crystal を選んで、余計な色の Lantern を取り除いて、必要な色の Lantern を置こうとする
    • 2色の Crystal は、1色だけ合ってる時は色無しの時よりスコアが悪いので、2色同時に揃えるようにすることで対策した。
      • ”Incorrect な Crystal を選んで、Lantern の位置を変えて、Correct になるまで”を1回の遷移とするSAも試したけど、調整不足だったのか、あまり Correct になってなかった。

高速化

tomerunさんの記事(北大日立新概念マラソンでやった高速化色々)がまとまってます。

  • 動的メモリは確保しない
    • 2倍高速化
    • 最初はメモリのことはあまり気にしなかったので、大変効果が有りました。
  • 除算・剰余算を取り除く
    • 観察する限り効果なし
  • スコア差分計算のために現在の値を保持
    • 試してないです
    • Crystal ごとに出来たかも
  • ビット並列化
    • 試してないです
  • メモリサイズを減らす
    • 試してないです
  • 座標を1次元に潰す
    • 20%高速化
  • 番兵で条件判定を取り除く
    • 観察する限り効果なし
  • 焼きなまし受理判定の枝刈り
    • 最初から使ってた
  • ループアンローリング
    • 試してないです
    • 4方向に ray を出してたので、使えました。
  • BFSで来た方向に戻る移動をしない
    • BFSを使っていない
  • 0〜N-1から異なる2つをランダムに選ぶときのテク
    • 異なる2つを選ぶ機会がなかった


source code

import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.util.ArrayList;
import java.util.Arrays;

public class CrystalLighting {
private IntSet currentRays = new IntSet(128 * 128 * 16);

private static final int blue = 1;
private static final int yellow = 2;
private static final int red = 4;
private static final int obstacle = 7;
private static final int empty = 0;
private static final int mirror1 = 8;
private static final int mirror2 = 9;
private static final int wall = 10;

private int[] objects;
private IntList[] rays;
private IntList updateRays = new IntList(128 * 128 * 4);

private IntSet emptySet;
private IntSet objectSet;
private IntSet crystalSet;
private IntSet incorrectCrystalSet;

private IntList bestObjects;

private int H;
private int W;
private static final int maxW = 128;
private int[] board;
private int costLantern;
private int costMirror;
private int costObstacle;
private int maxMirrors;
private int maxObstacles;

private static final int[] dz = { -128, -1, 1, 128, };

private int usedLantern;
private int usedObstacles;
private int usedMirrors;

private int numCorrects1;
private int numCorrects2;
private int numIncorrects;

private IntList conflictPoints = new IntList(2);
private IntList availableDirections = new IntList(4);

private IntList removes;
private IntList adds;
private IntList points;

private int[] typeLanterns = new int[] { blue, yellow, red, };
private int[] typeNotLanterns = new int[] { obstacle, mirror1, mirror2, };
private int[] typeObstacles = new int[] { obstacle, };
private int[] typeNotObstacles = new int[] { blue, yellow, red, mirror1, mirror2, };
private int[] typeMirrors = new int[] { mirror1, mirror2, };
private int[] typeNotMirrors = new int[] { blue, yellow, red, obstacle, };
private int[] types = new int[] { blue, yellow, red, obstacle, mirror1, mirror2, };

private double score;
private double bestScore;
static final Watch watch = new Watch();
static final XorShift rng = new XorShift(System.nanoTime());
private SAState sa = new SAState();

public String[] placeItems(String[] targetBoard, int costLantern, int costMirror, int costObstacle, int maxMirrors, int maxObstacles) {
init(targetBoard, costLantern, costMirror, costObstacle, maxMirrors, maxObstacles);
solve();
return makeResult();
}

private void init(String[] targetBoard, int costLantern, int costMirror, int costObstacle, int maxMirrors, int maxObstacles) {
initBoard(targetBoard, costLantern, costMirror, costObstacle, maxMirrors, maxObstacles);
Utils.debug("init", "H", H, "W", W, "costLantern", costLantern, "costMirror", costMirror, "costObstacle", costObstacle, "maxMirrors", maxMirrors, "maxObstacles", maxObstacles, "time", watch.getSecondString());
}

private void solve() {
SA();
}

private void SA() {
score = getScore();
bestScore = 0;
saveBest();

sa.startTime = watch.getSecond();
sa.endTime = 9.5;
sa.init();
for (;; sa.numIterations++) {
if ((sa.numIterations & ((1 << 10) - 1)) == 0) {
sa.update();

if (sa.isTLE()) {
loadBest();
break;
}
}

mutate();
}

Utils.debug("numIterations", sa.numIterations);
Utils.debug("SA", "time", watch.getSecondString());
printNumbers();
}

private void mutate() {
int random = (int) (6 * rng.nextDouble());
if (random < 1) {
remove();
} else if (random < 2) {
move();
} else if (random < 3) {
add();
} else if (random < 4) {
forceAdd();
} else if (random < 5) {
match3color();
} else if (random < 6) {
forceAdd();
}
}

private void add() {
if (emptySet.isEmpty()) {
return;
}

int z = emptySet.get((int) (emptySet.size() * rng.nextDouble()));
int type;
if (getRays(z).size() > 0) {
if (usedObstacles >= maxObstacles) {
if (usedMirrors >= maxMirrors) {
return;
} else {
type = typeMirrors[(int) (typeMirrors.length * rng.nextDouble())];
}
} else {
if (usedMirrors >= maxMirrors) {
type = typeObstacles[(int) (typeObstacles.length * rng.nextDouble())];
} else {
type = typeNotLanterns[(int) (typeNotLanterns.length * rng.nextDouble())];
}
}
} else {
type = typeLanterns[(int) (typeLanterns.length * rng.nextDouble())];
}

if (!canAdd(z, type)) {
return;
}

updateRays.clear();
double deltaScore = simulateAdd(z, type);
back();

if (sa.accept(deltaScore)) {
score += deltaScore;
add(z, type);
saveBest();
}
}

private void forceAdd() {
if (emptySet.isEmpty()) {
return;
}

int z = emptySet.get((int) (emptySet.size() * rng.nextDouble()));
assert getBoard(z) == empty && getObject(z) == empty;

if (getRays(z).size() == 0) {
int type = typeLanterns[(int) (typeLanterns.length * rng.nextDouble())];

if (!canAdd(z, type)) {
return;
}

updateRays.clear();
double deltaScore = simulateAdd(z, type);
back();

if (sa.accept(deltaScore)) {
score += deltaScore;
add(z, type);
saveBest();
}

return;
}

int type;
if (usedObstacles >= maxObstacles) {
if (usedMirrors >= maxMirrors) {
type = typeLanterns[(int) (typeLanterns.length * rng.nextDouble())];
} else {
type = typeNotObstacles[(int) (typeNotObstacles.length * rng.nextDouble())];
}
} else {
if (usedMirrors >= maxMirrors) {
type = typeNotMirrors[(int) (typeNotMirrors.length * rng.nextDouble())];
} else {
type = types[(int) (types.length * rng.nextDouble())];
}
}

if (type == blue || type == yellow || type == red) {
IntList conflicts = conflictPoints;
conflicts.clear();
for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
int encode = encode(z(ray), getObject(z(ray)));
if (!conflicts.contains(encode)) {
conflicts.add(encode);
}
}

for (int i = 0; i < conflicts.size(); i++) {
int conflict = conflicts.get(i);
remove(z(conflict));
}

if (!canAdd(z, type)) {
for (int i = 0; i < conflicts.size(); i++) {
int conflict = conflicts.get(i);
add(z(conflict), typeOrDirection(conflict));
}
return;
}

double deltaScore0 = getScore() - score;

updateRays.clear();
double deltaScore = deltaScore0 + simulateAdd(z, type);
back();

if (sa.accept(deltaScore)) {
score += deltaScore;
add(z, type);
saveBest();
} else {
for (int i = 0; i < conflicts.size(); i++) {
int conflict = conflicts.get(i);
add(z(conflict), typeOrDirection(conflict));
}
}
} else {
if (!canAdd(z, type)) {
return;
}

updateRays.clear();
double deltaScore = simulateAdd(z, type);
back();

if (sa.accept(deltaScore)) {
score += deltaScore;
add(z, type);
saveBest();
}
}

}

private void match3color() {
if (incorrectCrystalSet.isEmpty()) {
return;
}
int z = incorrectCrystalSet.get((int) (incorrectCrystalSet.size() * rng.nextDouble()));
assert !isCorrect(z);

if (availableDirections(z) < Integer.bitCount(getBoard(z))) {
return;
}

int color = getColor(z);
int crystalColor = getBoard(z);

removes.clear();
adds.clear();

points.clear();
for (int d = 0; d < dz.length; d++) {
search(z, d, points);
}
if (points.size() == 0) {
return;
}

for (int diffColor = color ^ crystalColor; diffColor > 0; diffColor -= Integer.lowestOneBit(diffColor)) {
int lowestColor = Integer.lowestOneBit(diffColor);
if ((crystalColor & lowestColor) == 0) {
for (int i = 0; i < 4; i++) {
if (getRays(z).size() == 0) {
break;
}
int ray = getRays(z).get((int) (getRays(z).size() * rng.nextDouble()));
int z2 = z(ray);
if (getObject(z2) == lowestColor) {
if (canRemove(z2)) {
int type = remove(z2);
removes.add(encode(z2, type));
break;
}
}
}
} else {
for (int i = 0; i < 3; i++) {
int p = points.get((int) (points.size() * rng.nextDouble()));
int z2 = p;
if (canAdd(z2, lowestColor)) {
add(z2, lowestColor);
adds.add(encode(z2, lowestColor));
break;
}
}
}
}

if (adds.size() == 0 && removes.size() == 0) {
return;
}

double deltaScore = getScore() - score;

if (sa.accept(deltaScore)) {
score += deltaScore;
saveBest();
} else {
for (int i = 0; i < adds.size(); i++) {
int add = adds.get(i);
remove(z(add));
}
for (int i = 0; i < removes.size(); i++) {
int remove = removes.get(i);
add(z(remove), typeOrDirection(remove));
}
}

}

private int availableDirections(int z) {
int count = 0;
for (int d = 0; d < dz.length; d++) {
int nz = z + dz[d];
if (isWall(nz)) {
continue;
}

if (getBoard(nz) == empty) {
count++;
}
}
return count;
}

private void move() {
if (objectSet.isEmpty()) {
return;
}

int z = objectSet.get((int) (objectSet.size() * rng.nextDouble()));

availableDirections.clear();
for (int d = 0; d < dz.length; d++) {
int newZ = z + dz[d];
if (isWall(newZ)) {
continue;
}
if (getBoard(newZ) != empty) {
continue;
}
if (getObject(newZ) != empty) {
continue;
}
availableDirections.add(d);
}
if (availableDirections.isEmpty()) {
return;
}

if (!canRemove(z)) {
return;
}

int type = remove(z);
double deltaScore0 = getScore() - score;

int d = availableDirections.get((int) (availableDirections.size() * rng.nextDouble()));
int newZ = z + dz[d];
if (!canAdd(newZ, type)) {
add(z, type);
return;
}

updateRays.clear();
double deltaScore = deltaScore0 + simulateAdd(newZ, type);
back();

if (sa.accept(deltaScore)) {
score += deltaScore;
add(newZ, type);
saveBest();
} else {
add(z, type);
}
}

private void search(int z0, int d0, IntList points) {
int d = d0;
int z = z0;
for (;;) {
z += dz[d];
if (isWall(z)) {
break;
}

if (z == z0) {
break;
}
if (isLantern(z)) {
continue;
}
if (isCrystal(z)) {
break;
}
if (isObstacle(z)) {
break;
}
if (getObject(z) == mirror1) {
d ^= 1;
continue;
} else if (getObject(z) == mirror2) {
d ^= 2;
continue;
}

points.add(z);
}
}

private boolean isCorrect(int z) {
assert isCrystal(z);
return getColor(z) == getBoard(z);
}

private int getColor(int z) {
int color = 0;
for (int i = 0; i < getRays(z).size(); i++) {
color |= getObject(z(getRays(z).get(i)));
}
return color;
}

private void remove() {
if (objectSet.isEmpty()) {
return;
}

int z = objectSet.get((int) (objectSet.size() * rng.nextDouble()));

if (!canRemove(z)) {
return;
}

updateRays.clear();
double deltaScore = simulateRemove(z);
back();

if (sa.accept(deltaScore)) {
score += deltaScore;
remove(z);
saveBest();
}
}

private void back() {
for (int i = updateRays.size() - 1; i >= 0; i -= 3) {
int v3 = updateRays.get(i);
int v2 = updateRays.get(i - 1);
int v = updateRays.get(i - 2);
if (v > 0) {
getRays(v2).remove(getRays(v2).indexOf(v3));
} else {
getRays(v2).add(v3);
}
}
}

private void saveBest() {
if (score > bestScore) {
bestScore = score;
bestObjects.clear();
for (int i = 0; i < objectSet.size(); i++) {
int z = objectSet.get(i);
bestObjects.add(encode(z, getObject(z)));
}
}
}

private void loadBest() {
score = bestScore;

for (int i = objectSet.size() - 1; i >= 0; i--) {
int z = objectSet.get(i);
if (isLantern(z)) {
remove(z);
}
}
for (int i = objectSet.size() - 1; i >= 0; i--) {
int z = objectSet.get(i);
if (!isLantern(z)) {
remove(z);
}
}

for (int i = 0; i < bestObjects.size(); i++) {
int v = bestObjects.get(i);
int z = z(v);
int type = typeOrDirection(v);

if (type == blue || type == yellow || type == red) {
} else {
add(z, type);
}
}
for (int i = 0; i < bestObjects.size(); i++) {
int v = bestObjects.get(i);
int z = z(v);
int type = typeOrDirection(v);

if (type == blue || type == yellow || type == red) {
add(z, type);
} else {
}
}
}

private void initBoard(String[] targetBoard, int costLantern, int costMirror, int costObstacle, int maxMirrors, int maxObstacles) {
this.H = targetBoard.length;
this.W = targetBoard[0].length();
this.board = new int[(H + 2) * maxW];
Arrays.fill(this.board, wall);
for (int r = 0; r < H; r++) {
for (int c = 0; c < W; c++) {
char charAt = targetBoard[r].charAt(c);
int z = getZ(r + 1, c + 1);
if (charAt == '1') {
setBoard(z, blue);
} else if (charAt == '2') {
setBoard(z, yellow);
} else if (charAt == '3') {
setBoard(z, blue + yellow);
} else if (charAt == '4') {
setBoard(z, red);
} else if (charAt == '5') {
setBoard(z, blue + red);
} else if (charAt == '6') {
setBoard(z, yellow + red);
} else if (charAt == 'X') {
setBoard(z, obstacle);
} else if (charAt == '.') {
setBoard(z, empty);
}
}
}

objects = new int[(H + 2) * maxW];
rays = new IntList[(H + 2) * maxW];
for (int i = 0; i < rays.length; i++) {
rays[i] = new IntList(4);
}
this.costLantern = -costLantern;
this.costMirror = -costMirror;
this.costObstacle = -costObstacle;
this.maxMirrors = maxMirrors;
this.maxObstacles = maxObstacles;

emptySet = new IntSet((H + 2) * maxW);
for (int r = 0; r < H; r++) {
for (int c = 0; c < W; c++) {
int z = getZ(r + 1, c + 1);
if (getBoard(z) == empty) {
emptySet.add(z);
}
}
}

objectSet = new IntSet((H + 2) * maxW);

crystalSet = new IntSet((H + 2) * maxW);
incorrectCrystalSet = new IntSet((H + 2) * maxW);
for (int r = 0; r < H; r++) {
for (int c = 0; c < W; c++) {
int z = getZ(r + 1, c + 1);
if (isCrystal(z)) {
crystalSet.add(z);
incorrectCrystalSet.add(z);
}
}
}

bestObjects = new IntList((H + 2) * maxW);

removes = new IntList((H + 2) * maxW);
adds = new IntList((H + 2) * maxW);
points = new IntList((H + 2) * maxW);

}

private int getBoard(int z) {
return board[z];
}

private int setBoard(int z, int v) {
return board[z] = v;
}

private IntList getRays(int z) {
return rays[z];
}

private int getObject(int z) {
return objects[z];
}

private void setObject(int z, int v) {
objects[z] = v;
}

private int getScore() {
return numCorrects1 * 20 + numCorrects2 * 30 + numIncorrects * -10 + usedLantern * costLantern + usedObstacles * costObstacle + usedMirrors * costMirror;
}

private boolean canAdd(int z, int type) {
if (!(getBoard(z) == empty && getObject(z) == empty)) {
return false;
}
if (type == obstacle) {
if (usedObstacles >= maxObstacles) {
return false;
}
assert getObject(z) == empty;
setObject(z, type);
for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
if (!isValidRay(z(ray), typeOrDirection(ray))) {
setObject(z, empty);
return false;
}
}
setObject(z, empty);
return true;
} else if (type == mirror1 || type == mirror2) {
if (usedMirrors >= maxMirrors) {
return false;
}
assert getObject(z) == empty;
setObject(z, type);
for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
if (!isValidRay(z(ray), typeOrDirection(ray))) {
setObject(z, empty);
return false;
}
}
setObject(z, empty);
return true;
} else if (type == blue || type == yellow || type == red) {
if (getRays(z).size() > 0) {
return false;
}

assert getObject(z) == empty;
setObject(z, type);

for (int d = 0; d < dz.length; d++) {
if (!isValidRay(z, d)) {
setObject(z, empty);
return false;
}
}
setObject(z, empty);
return true;
}
throw new AssertionError();
}

private boolean canRemove(int z) throws AssertionError {
int type = getObject(z);
if (type == empty) {
return false;
}
if (type == blue || type == yellow || type == red) {
return true;
} else if (type == obstacle || type == mirror1 || type == mirror2) {
setObject(z, empty);
for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
if (!isValidRay(z(ray), typeOrDirection(ray))) {
setObject(z, type);
return false;
}
}
setObject(z, type);
return true;
}
throw new AssertionError();
}

private void add(int z, int type) throws AssertionError {
if (type == blue || type == yellow || type == red) {
setObject(z, type);
emptySet.removeValue(z);
objectSet.add(z);
usedLantern++;

for (int d = 0; d < dz.length; d++) {
addRay(z, d);
}
} else if (type == obstacle || type == mirror1 || type == mirror2) {
currentRays.clear();
for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
currentRays.add(ray);
}

for (int i = 0; i < currentRays.size(); i++) {
int ray = currentRays.get(i);
removeRay(z(ray), typeOrDirection(ray));
}

setObject(z, type);
emptySet.removeValue(z);
objectSet.add(z);
if (type == obstacle) {
usedObstacles++;
} else {
usedMirrors++;
}

for (; currentRays.size() > 0;) {
int ray = currentRays.get(0);
currentRays.remove(0);
addRay(z(ray), typeOrDirection(ray));
}
} else {
throw new AssertionError(Utils.toString("type", type));
}
}

private int remove(int z) throws AssertionError {
int type = getObject(z);
if (type == blue || type == yellow || type == red) {
for (int d = 0; d < dz.length; d++) {
removeRay(z, d);
}

setObject(z, empty);
emptySet.add(z);
objectSet.removeValue(z);
usedLantern--;
} else if (type == obstacle || type == mirror1 || type == mirror2) {
currentRays.clear();
for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
currentRays.add(ray);
}

for (int i = 0; i < currentRays.size(); i++) {
int ray = currentRays.get(i);
removeRay(z(ray), typeOrDirection(ray));
}

setObject(z, empty);
emptySet.add(z);
objectSet.removeValue(z);
if (type == obstacle) {
usedObstacles--;
} else {
usedMirrors--;
}

for (; currentRays.size() > 0;) {
int ray = currentRays.get(0);
currentRays.remove(0);
addRay(z(ray), typeOrDirection(ray));
}

} else {
throw new AssertionError(Utils.toString("type", type));
}
return type;
}

private int simulateAdd(int z, int type) throws AssertionError {
if (type == blue || type == yellow || type == red) {
int deltaScore = 0;

assert getObject(z) == empty;

setObject(z, type);

deltaScore += costLantern;
for (int d = 0; d < dz.length; d++) {
deltaScore += simulateAddRay2(z, d);
}

setObject(z, empty);

return deltaScore;
} else if (type == obstacle || type == mirror1 || type == mirror2) {
int deltaScore = 0;

assert getObject(z) == empty;

currentRays.clear();
for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
currentRays.add(ray);
}

for (int i = 0; i < currentRays.size(); i++) {
int ray = currentRays.get(i);
deltaScore += simulateRemoveRay2(z(ray), typeOrDirection(ray));
}

setObject(z, type);

if (type == obstacle) {
deltaScore += costObstacle;
} else {
deltaScore += costMirror;
}

for (; currentRays.size() > 0;) {
int ray = currentRays.get(0);
currentRays.remove(0);
deltaScore += simulateAddRay2(z(ray), typeOrDirection(ray));
}

setObject(z, empty);

return deltaScore;
} else {
throw new AssertionError(Utils.toString("type", type));
}
}

private int simulateRemove(int z) throws AssertionError {
int type = getObject(z);
if (type == blue || type == yellow || type == red) {
int deltaScore = 0;

for (int d = 0; d < dz.length; d++) {
deltaScore += simulateRemoveRay2(z, d);
}

deltaScore += -costLantern;

return deltaScore;
} else if (type == obstacle || type == mirror1 || type == mirror2) {
int deltaScore = 0;

currentRays.clear();
for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
currentRays.add(ray);
}

for (int i = 0; i < currentRays.size(); i++) {
int ray = currentRays.get(i);
deltaScore += simulateRemoveRay2(z(ray), typeOrDirection(ray));
}

setObject(z, empty);

if (type == obstacle) {
deltaScore += -costObstacle;
} else {
deltaScore += -costMirror;
}

for (; currentRays.size() > 0;) {
int ray = currentRays.get(0);
currentRays.remove(0);
deltaScore += simulateAddRay2(z(ray), typeOrDirection(ray));
}

setObject(z, type);

return deltaScore;
} else {
throw new AssertionError(Utils.toString("type", type));
}
}

private boolean isValidRay(int z0, int d0) {
int d = d0;
int z = z0;
for (;;) {
z += dz[d];
if (isWall(z)) {
break;
}
if (z == z0) {
return false;
}
if (isLantern(z)) {
return false;
}
if (isCrystal(z)) {
break;
}
if (isObstacle(z)) {
break;
}
if (getObject(z) == mirror1) {
d ^= 1;
} else if (getObject(z) == mirror2) {
d ^= 2;
}
}
return true;
}

private void addRay(int z0, int d0) {
int value = encode(z0, d0);
assert value == encode(getR(z0), getC(z0), d0);
int d = d0;
int z = z0;
for (;;) {
z += dz[d];
if (isWall(z)) {
break;
}

assert !isLantern(z);

getRays(z).add(value);
if (isCrystal(z)) {
int color = 0;
for (int i = 0; i < getRays(z).size() - 1; i++) {
int ray = getRays(z).get(i);
color |= getObject(z(ray));
}

int bitCount = Integer.bitCount(getBoard(z));
if (color == getBoard(z)) {
if (bitCount == 1) {
numCorrects1--;
} else {
numCorrects2--;
}
} else {
if (color == 0) {
} else {
numIncorrects--;
}
}

{
int ray = getRays(z).get(getRays(z).size() - 1);
color |= getObject(z(ray));
}

if (color == getBoard(z)) {
if (bitCount == 1) {
numCorrects1++;
} else {
numCorrects2++;
}
incorrectCrystalSet.removeValue(z);
} else {
if (color == 0) {
} else {
numIncorrects++;
}
incorrectCrystalSet.add(z);
}
break;
}
if (isObstacle(z)) {
break;
}
if (getObject(z) == mirror1) {
d ^= 1;
} else if (getObject(z) == mirror2) {
d ^= 2;
}

}
}

private int simulateAddRay2(int z0, int d0) {
int deltaScore = 0;
int value = encode(z0, d0);
assert value == encode(getR(z0), getC(z0), d0);
int d = d0;
int z = z0;
for (;;) {
z += dz[d];
if (isWall(z)) {
break;
}

assert !isLantern(z);
if (isCrystal(z)) {
getRays(z).add(value);
updateRays.add(1);
updateRays.add(z);
updateRays.add(value);

int color = 0;
for (int i = 0; i < getRays(z).size() - 1; i++) {
int ray = getRays(z).get(i);
color |= getObject(z(ray));
}

int bitCount = Integer.bitCount(getBoard(z));
if (color == getBoard(z)) {
if (bitCount == 1) {
deltaScore += -20;
} else {
deltaScore += -30;
}
} else {
if (color == 0) {
} else {
deltaScore += 10;
}
}

{
assert z0 == z(value);
color |= getObject(z0);
}

if (color == getBoard(z)) {
if (bitCount == 1) {
deltaScore += 20;
} else {
deltaScore += 30;
}
} else {
if (color == 0) {
} else {
deltaScore += -10;
}
}

break;
}
if (isObstacle(z)) {
break;
}
if (getObject(z) == mirror1) {
d ^= 1;
} else if (getObject(z) == mirror2) {
d ^= 2;
}

}
return deltaScore;
}

private void removeRay(int z0, int d0) {
int value = encode(z0, d0);
int d = d0;
int z = z0;
for (;;) {
z += dz[d];
if (isWall(z)) {
break;
}

assert !isLantern(z) : Utils.toString("nr, nc", getR(z), getC(z));

int indexOf = getRays(z).indexOf(value);
getRays(z).removeFast(indexOf);
if (isCrystal(z)) {

int color = 0;
for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
color |= getObject(z(ray));
}

int bitCount = Integer.bitCount(getBoard(z));
if (color == getBoard(z)) {
if (bitCount == 1) {
numCorrects1++;
} else {
numCorrects2++;
}
incorrectCrystalSet.removeValue(z);
} else {
if (color == 0) {
} else {
numIncorrects++;
}
incorrectCrystalSet.add(z);
}

{
color |= getObject(z0);
}

if (color == getBoard(z)) {
if (bitCount == 1) {
numCorrects1--;
} else {
numCorrects2--;
}
} else {
if (color == 0) {
} else {
numIncorrects--;
}
}

break;
}
if (isObstacle(z)) {
break;
}
if (getObject(z) == mirror1) {
d ^= 1;
} else if (getObject(z) == mirror2) {
d ^= 2;
}

}
}

private int simulateRemoveRay2(int z0, int d0) {
int deltaScore = 0;
int value = encode(z0, d0);
int d = d0;
int z = z0;
for (;;) {
z += dz[d];
if (isWall(z)) {
break;
}

assert !isLantern(z) : Utils.toString("nr, nc", getR(z), getC(z));
if (isCrystal(z)) {
int indexOf = getRays(z).indexOf(value);
int removedRay = getRays(z).removeFast(indexOf);
updateRays.add(-1);
updateRays.add(z);
updateRays.add(removedRay);

int color = 0;

for (int i = 0; i < getRays(z).size(); i++) {
int ray = getRays(z).get(i);
color |= getObject(z(ray));
}

int bitCount = Integer.bitCount(getBoard(z));
if (color == getBoard(z)) {
if (bitCount == 1) {
deltaScore += 20;
} else {
deltaScore += 30;
}
} else {
if (color == 0) {
} else {
deltaScore += -10;
}
}

{
color |= getObject(z0);
}

if (color == getBoard(z)) {
if (bitCount == 1) {
deltaScore += -20;
} else {
deltaScore += -30;
}
} else {
if (color == 0) {
} else {
deltaScore += 10;
}
}

break;
}
if (isObstacle(z)) {
break;
}
if (getObject(z) == mirror1) {
d ^= 1;
} else if (getObject(z) == mirror2) {
d ^= 2;
}

}
return deltaScore;
}

private boolean isWall(int z) {
return getBoard(z) == wall;
}

private boolean isObstacle(int z) {
return getBoard(z) == obstacle || getObject(z) == obstacle;
}

private boolean isCrystal(int z) {
return getBoard(z) >= 1 && getBoard(z) <= 6;
}

private boolean isLantern(int z) {
return getObject(z) == blue || getObject(z) == yellow || getObject(z) == red;
}

private static final int getZ(int r, int c) {
return (r << 7) | c;
}

private static final int getR(int z) {
return z >>> 7;
}

private static final int getC(int z) {
return z & 127;
}

private static final int encode(int r, int c, int type) {
return (r << 11) | (c << 4) | (type);
}

private static final int encode(int z, int type) {
return (z << 4) | (type);
}

private static final int z(int v) {
return v >>> 4;
}

private static final int typeOrDirection(int v) {
return v & 15;
}

private void printNumbers() {
Utils.debug("score", getScore(), "Lan", usedLantern, "Obs", usedObstacles, "Mir", usedMirrors, "Co1", numCorrects1, "Co2", numCorrects2, "Inc", numIncorrects);
}

private String[] makeResult() {
ArrayList<String> res = new ArrayList<>();
for (int r = 0; r < H; r++) {
for (int c = 0; c < W; c++) {
int z = getZ(r + 1, c + 1);
if (getObject(z) != empty) {
char ch;
if (getObject(z) == blue || getObject(z) == yellow || getObject(z) == red) {
ch = (char) (getObject(z) + '0');
} else if (getObject(z) == obstacle) {
ch = 'X';
} else if (getObject(z) == mirror1) {
ch = '\\';
} else if (getObject(z) == mirror2) {
ch = '/';
} else {
throw new AssertionError();
}
res.add("" + r + " " + c + " " + ch);
}
}
}

return (String[]) res.toArray(new String[res.size()]);
}

public static void main(String[] args) {
try (BufferedReader br = new BufferedReader(new InputStreamReader(System.in))) {

int H = Integer.parseInt(br.readLine());
String[] targetBoard = new String[H];
for (int i = 0; i < H; ++i) {
targetBoard[i] = br.readLine();
}
int costLantern = Integer.parseInt(br.readLine());
int costMirror = Integer.parseInt(br.readLine());
int costObstacle = Integer.parseInt(br.readLine());

int maxMirrors = Integer.parseInt(br.readLine());
int maxObstacles = Integer.parseInt(br.readLine());

CrystalLighting cl = new CrystalLighting();
String[] ret = cl.placeItems(targetBoard, costLantern, costMirror, costObstacle, maxMirrors, maxObstacles);
System.out.println(ret.length);
for (int i = 0; i < ret.length; ++i) {
System.out.println(ret[i]);
}
System.out.flush();
} catch (Exception e) {
e.printStackTrace();
}
}

}

class SAState {

public static final boolean useTime = true;

public double startTime = 0;
public double endTime = 9.5;
public double time = startTime;

public double startTemperature = 10;
public double endTemperature = 1.25;
public double inverseTemperature = 1.0 / startTemperature;
public double startRange = 31;
public double endRange = 3;
public double range = startRange;

public int numIterations;

public void init() {
numIterations = 0;
startTime = useTime ? CrystalLighting.watch.getSecond() : numIterations;

update();
}

public void update() {
updateTime();
updateTemperature();
updateRange();
}

public void updateTemperature() {
inverseTemperature = 1.0 / (endTemperature + (startTemperature - endTemperature) * Math.pow((endTime - time) / (endTime - startTime), 1.0));
}

public void updateRange() {
range = endRange + (startRange - endRange) * Math.pow((endTime - time) / (endTime - startTime), 1.0);
}

public void updateTime() {
time = useTime ? CrystalLighting.watch.getSecond() : numIterations;
}

public boolean isTLE() {
return time >= endTime;
}

public boolean accept(double deltaScore) {
return acceptB(deltaScore);
}

public boolean acceptB(double deltaScore) {
if (deltaScore > -1e-9) {
return true;
}

assert deltaScore < 0;
assert 1.0 / inverseTemperature >= 0;

if (deltaScore * inverseTemperature < -10) {
return false;
}

if (CrystalLighting.rng.nextDouble() < Math.exp(deltaScore * inverseTemperature)) {
return true;
}
return false;
}

public boolean acceptS(double deltaScore) {
if (deltaScore < 1e-9) {
return true;
}

assert deltaScore > 0;
assert 1.0 / inverseTemperature >= 0;

if (-deltaScore * inverseTemperature < -10) {
return false;
}

if (CrystalLighting.rng.nextDouble() < Math.exp(-deltaScore * inverseTemperature)) {
return true;
}
return false;
}

}

final class Utils {
private Utils() {
}

public static final void debug(Object... o) {
System.err.println(toString(o));
}

public static final String toString(Object... o) {
return Arrays.deepToString(o);
}

public static final boolean isValid(int v, int min, int minUpper) {
return v >= min && v < minUpper;
}

}

class Watch {
private long start;

public Watch() {
init();
}

public double getSecond() {
return (System.nanoTime() - start) * 1e-9;
}

public void init() {
init(System.nanoTime());
}

private void init(long start) {
this.start = start;
}

public String getSecondString() {
return toString(getSecond());
}

public static final String toString(double second) {
if (second < 60) {
return String.format("%5.2fs", second);
} else if (second < 60 * 60) {
int minute = (int) (second / 60);
return String.format("%2dm%2ds", minute, (int) (second % 60));
} else {
int hour = (int) (second / (60 * 60));
int minute = (int) (second / 60);
return String.format("%2dh%2dm%2ds", hour, minute % (60), (int) (second % 60));
}
}

}

class XorShift {
private int w = 88675123;
private int x = 123456789;
private int y = 362436069;
private int z = 521288629;

public XorShift(long l) {
x = (int) l;
}

public int nextInt() {
final int t = x ^ (x << 11);
x = y;
y = z;
z = w;
w = w ^ (w >>> 19) ^ (t ^ (t >>> 8));
return w;
}

public long nextLong() {
return ((long) nextInt() << 32) ^ (long) nextInt();
}

public double nextDouble() {
return (nextInt() >>> 1) * 4.6566128730773926E-10;
}

public int nextInt(int n) {
return (int) (n * nextDouble());
}

}

class IntSet {
private static final int EMPTY = -1;
private int size;
private int[] indexToValue;
private int[] valueToIndex;

public IntSet(int capacity) {
this.size = 0;
indexToValue = new int[capacity];
valueToIndex = new int[capacity];
Arrays.fill(valueToIndex, EMPTY);
}

public boolean add(int value) {
if (valueToIndex[value] != EMPTY) {
return false;
}
indexToValue[size] = value;
valueToIndex[indexToValue[size]] = size;
size++;
return true;
}

public boolean remove(int index) {
if (size == 0) {
return false;
}
assert index < size;
int swap = indexToValue[index];
indexToValue[index] = indexToValue[size - 1];
indexToValue[size - 1] = swap;

valueToIndex[indexToValue[index]] = index;
valueToIndex[indexToValue[size - 1]] = EMPTY;

size--;
return true;
}

public boolean removeValue(int value) {
int index = indexOf(value);
if (index == EMPTY) {
return false;
}
remove(index);
return true;
}

public int get(int index) {
assert index < size;
return indexToValue[index];
}

public int indexOf(int value) {
return valueToIndex[value];
}

public int size() {
return size;
}

public boolean isEmpty() {
return size() <= 0;
}

public void clear() {
for (; size() > 0;) {
remove(0);
}
}

public boolean contains(int value) {
return indexOf(value) != EMPTY;
}

}

class IntList {
private static final int EMPTY = -1;
private int[] values;
private int size;

public IntList(int capacity) {
values = new int[capacity];
clear();
}

public void clear() {
size = 0;
}

public int size() {
return size;
}

public boolean isEmpty() {
return size() <= 0;
}

public boolean contains(int value) {
return indexOf(value) != EMPTY;
}

public boolean add(int value) {
values[size++] = value;
return true;
}

public int remove(int index) {
int value = values[index];
for (int i = index; i + 1 < size; i++) {
values[i] = values[i + 1];
}
size--;
return value;
}

public int removeFast(int index) {
int value = values[index];
values[index] = values[size - 1];
size--;
return value;

}

public int get(int index) {
return values[index];
}

public int set(int index, int value) {
int oldValue = values[index];
values[index] = value;
return oldValue;
}

public void add(int index, int value) {
assert index <= size;
for (int i = size - 1; i >= index; i--) {
values[i + 1] = values[i];
}
size++;
values[index] = value;
}

public int indexOf(int value) {
for (int i = 0; i < size; i++) {
if (values[i] == value) {
return i;
}
}
return EMPTY;
}

public int lastIndexOf(int value) {
for (int i = size - 1; i >= 0; i--) {
if (values[i] == value) {
return i;
}
}
return EMPTY;
}

}