Problem:
Score : 500 points

Problem Statement

"Teishi-zushi", a Japanese restaurant, is a plain restaurant with only one round counter. The outer circumference of the counter is C meters. Customers cannot go inside the counter.

Nakahashi entered Teishi-zushi, and he was guided to the counter. Now, there are N pieces of sushi (vinegared rice with seafood and so on) on the counter. The distance measured clockwise from the point where Nakahashi is standing to the point where the i-th sushi is placed, is x_i meters. Also, the i-th sushi has a nutritive value of v_i kilocalories.

Nakahashi can freely walk around the circumference of the counter. When he reach a point where a sushi is placed, he can eat that sushi and take in its nutrition (naturally, the sushi disappears). However, while walking, he consumes 1 kilocalories per meter.

Whenever he is satisfied, he can leave the restaurant from any place (he does not have to return to the initial place). On balance, at most how much nutrition can he take in before he leaves? That is, what is the maximum possible value of the total nutrition taken in minus the total energy consumed? Assume that there are no other customers, and no new sushi will be added to the counter. Also, since Nakahashi has plenty of nutrition in his body, assume that no matter how much he walks and consumes energy, he never dies from hunger.

Constraints

1 ≤ N ≤ 10^5

2 ≤ C ≤ 10^{14}

1 ≤ x_1 < x_2 < ... < x_N < C

1 ≤ v_i ≤ 10^9

All values in input are integers.

Subscores

300 points will be awarded for passing the test set satisfying N ≤ 100.

Input

Input is given from Standard Input in the following format:

N C
x_1 v_1
x_2 v_2
:
x_N v_N

Output

If Nakahashi can take in at most c kilocalories on balance before he leaves the restaurant, print c.

Sample Input 1

3 20
2 80
9 120
16 1

Sample Output 1

191

There are three sushi on the counter with a circumference of 20 meters. If he walks two meters clockwise from the initial place, he can eat a sushi of 80 kilocalories. If he walks seven more meters clockwise, he can eat a sushi of 120 kilocalories. If he leaves now, the total nutrition taken in is 200 kilocalories, and the total energy consumed is 9 kilocalories, thus he can take in 191 kilocalories on balance, which is the largest possible value.

Sample Input 2

3 20
2 80
9 1
16 120

Sample Output 2

192

The second and third sushi have been swapped. Again, if he walks two meters clockwise from the initial place, he can eat a sushi of 80 kilocalories. If he walks six more meters counterclockwise this time, he can eat a sushi of 120 kilocalories. If he leaves now, the total nutrition taken in is 200 kilocalories, and the total energy consumed is 8 kilocalories, thus he can take in 192 kilocalories on balance, which is the largest possible value.

Sample Input 3

1 100000000000000
50000000000000 1

Sample Output 3

0

Even though the only sushi is so far that it does not fit into a 32-bit integer, its nutritive value is low, thus he should immediately leave without doing anything.

Sample Input 4

15 10000000000
400000000 1000000000
800000000 1000000000
1900000000 1000000000
2400000000 1000000000
2900000000 1000000000
3300000000 1000000000
3700000000 1000000000
3800000000 1000000000
4000000000 1000000000
4100000000 1000000000
5200000000 1000000000
6600000000 1000000000
8000000000 1000000000
9300000000 1000000000
9700000000 1000000000

Sample Output 4

6500000000

All these sample inputs above are included in the test set for the partial score.

Solution in C (GCC 5.4.1) (judge verdict: Accepted)

#include <limits.h>
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <time.h>

// 内部定数
#define D_SUSHI_MAX		100000									// 最大寿司数

// 内部構造体 - 寿司情報
typedef struct Sushi {
	long long mlLen;											// 距離
	long long mlCal;											// カロリー
	long long mlCSum;											// カロリー累計
} Sushi;

// 内部変数
static Sushi sz1SushiR[D_SUSHI_MAX];							// 寿司リスト(右側方向累計)
static Sushi sz1SushiL[D_SUSHI_MAX];							// 寿司リスト(左側方向累計)
static int siSCnt;												// 寿司リスト数
static long long slLen;											// 外周長さ
static long long slMax;											// 最大カロリー累計

// 内部変数 - テスト用
#ifdef D_TEST
	static int siRes;
	static FILE *szpFpT, *szpFpA;
#endif

// 寿司リスト - 累計セット・片側移動の最大値更新
int
fUpMaxSgl(
	Sushi *pzpSushi				// <I> 寿司リスト
)
{
	int i;

	// １つ目
	long long llLSum = pzpSushi[0].mlLen;
	long long llCSum = pzpSushi[0].mlCal - pzpSushi[0].mlLen;
	pzpSushi[0].mlCSum = llCSum;

	// 最大値更新
	if (slMax < llCSum) {
		slMax = llCSum;
	}

	// ２つ目以降
	for (i = 1; i < siSCnt; i++) {
		llLSum += pzpSushi[i].mlLen;
		llCSum += pzpSushi[i].mlCal - pzpSushi[i].mlLen + pzpSushi[i - 1].mlLen;
		if (llCSum < pzpSushi[i - 1].mlCSum) {										// 前回の寿司時点の方が高カロリーの場合
			pzpSushi[i].mlCSum = pzpSushi[i - 1].mlCSum;
		}
		else {																		// 今回の寿司が高カロリー
			pzpSushi[i].mlCSum = llCSum;

			// 最大値更新
			if (slMax < llCSum) {
				slMax = llCSum;
			}
		}
	}

	return 0;
}

// 寿司リスト - 両側移動の最大値更新
int
fUpMaxDbl(
	Sushi *pzpSushi1			// <I> 寿司リスト１
	, Sushi *pzpSushi2			// <I> 寿司リスト２
)
{
	int i;

	for (i = 0; i < siSCnt - 1; i++) {

		// リスト１方向へ
		long long llCSum = pzpSushi1[i].mlCSum;

		// 初期位置へ
		llCSum -= pzpSushi1[i].mlLen;

		// リスト２方向へ
		llCSum += pzpSushi2[siSCnt - i - 2].mlCSum;

		// 最大値更新
		if (slMax < llCSum) {
			slMax = llCSum;
		}
	}

	return 0;
}

// 実行メイン
int
fMain(
	int piTNo					// <I> テスト番号 1～
)
{
	int i;
	char lc1Buf[1024], lc1Out[1024];

	// データ初期化
	slMax = 0;													// 最大カロリー累計

	// テストファイルオープン
#ifdef D_TEST
	sprintf(lc1Buf, ".\\Test\\T%d.txt", piTNo);
	szpFpT = fopen(lc1Buf, "r");
	sprintf(lc1Buf, ".\\Test\\A%d.txt", piTNo);
	szpFpA = fopen(lc1Buf, "r");
	siRes = 0;
#endif

	// 寿司リスト数・外周長さ取得
#ifdef D_TEST
	fgets(lc1Buf, sizeof(lc1Buf), szpFpT);
#else
	fgets(lc1Buf, sizeof(lc1Buf), stdin);
#endif
	sscanf(lc1Buf, "%d%lld", &siSCnt, &slLen);

	// 寿司リスト(右側方向)取得
	for (i = 0; i < siSCnt; i++) {
#ifdef D_TEST
		fgets(lc1Buf, sizeof(lc1Buf), szpFpT);
#else
		fgets(lc1Buf, sizeof(lc1Buf), stdin);
#endif
		sscanf(lc1Buf, "%lld%lld", &sz1SushiR[i].mlLen, &sz1SushiR[i].mlCal);
	}

	// 寿司リスト(左側方向)セット
	for (i = 0; i < siSCnt; i++) {
		sz1SushiL[i].mlLen = slLen - sz1SushiR[siSCnt - i - 1].mlLen;
		sz1SushiL[i].mlCal = sz1SushiR[siSCnt - i - 1].mlCal;
	}

	// 寿司リスト - 累計セット・片側移動の最大値更新
	fUpMaxSgl(sz1SushiR);
	fUpMaxSgl(sz1SushiL);

	// 寿司リスト - 両側移動の最大値更新
	fUpMaxDbl(sz1SushiR, sz1SushiL);			// 右側方向＋左側方向
	fUpMaxDbl(sz1SushiL, sz1SushiR);			// 左側方向＋右側方向

	// 結果セット
	sprintf(lc1Out, "%lld\n", slMax);

	// 結果表示
#ifdef D_TEST
	fgets(lc1Buf, sizeof(lc1Buf), szpFpA);
	if (strcmp(lc1Buf, lc1Out)) {
		siRes = -1;
	}
#else
	printf("%s", lc1Out);
#endif

	// テストファイルクローズ
#ifdef D_TEST
	fclose(szpFpT);
	fclose(szpFpA);
#endif

	// テスト結果
#ifdef D_TEST
	if (siRes == 0) {
		printf("OK %d\n", piTNo);
	}
	else {
		printf("NG %d\n", piTNo);
	}
#endif

	return 0;
}

int
main()
{

#ifdef D_TEST
	int i;
	for (i = D_TEST_SNO; i <= D_TEST_ENO; i++) {
		fMain(i);
	}
#else
	fMain(0);
#endif

	return 0;
}