# Straight-forward 9ms 7-line c++ solution with explanation

• Search for the last stone in a depth-first way, prune those exceeding the [k-1,k+1] range. Well, I think the code is simple enough and need no more explanation.

``````bool canCross(vector<int>& stones, int pos = 0, int k = 0) {
for (int i = pos + 1; i < stones.size(); i++) {
int gap = stones[i] - stones[pos];
if (gap < k - 1) continue;
if (gap > k + 1) return false;
if (canCross(stones, i, gap)) return true;
}
return pos == stones.size() - 1;
}
``````

This can pass OJ at 9ms but is inefficient for extreme cases. (update: new test cases are added and the solution above no longer passes OJ, please see the solution below which takes 62ms) We can memorize the returns with minimum effort:

``````unordered_map<int, bool> dp;

bool canCross(vector<int>& stones, int pos = 0, int k = 0) {
int key = pos | k << 11;

if (dp.count(key) > 0)
return dp[key];

for (int i = pos + 1; i < stones.size(); i++) {
int gap = stones[i] - stones[pos];
if (gap < k - 1)
continue;
if (gap > k + 1)
return dp[key] = false;
if (canCross(stones, i, gap))
return dp[key] = true;
}

return dp[key] = (pos == stones.size() - 1);
}
``````

The number of stones is less than 1100 so pos will always be less than 2^11 (2048).
Stone positions could be theoretically up to 2^31 but k is practically not possible to be that big for the parameter as the steps must start from 0 and 1 and at the 1100th step the greatest valid k would be 1100. So combining pos and k is safe here.

``````  if (gap < k - 1) continue;
if (gap > k + 1) return false;
``````

that is, why u can safely prune like this??

• @HanMing.py
Because the stones are given ascendingly.

• Its time complexity is exponential.
In this case below, it will TLE:

`[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,99999999]`

• I also came up with the same solution and it passed OJ. However I don't think if this is the most efficient solution. The test cases do not cover a wide range of possibilities. If a test case involves a lot of back tracking then it might not be efficient.

What do you think is the time complexity of this solution ?

Thanks for you comments. I have updated the post.

• ``````int key = pos | k << 11;
``````

Why do you set the key this way?

• Wow I see it now. So you put both pos and k into the key for hashtable. It is the first time I see something like this. Very cool!

• @tonygogogo I think @mzchen just wanna map 'pos' and 'k' to a unique number, since we can not determine whether the last node is reachable without the parameter 'k'.

• @Abu-Q how can you guarantee the key is unique by this way? I cannot get it. Please refer to http://stackoverflow.com/questions/919612/mapping-two-integers-to-one-in-a-unique-and-deterministic-way to produce the unique key by two integer

• I think `int key = pos | k << 11;` will fail when we have a lot of stones in the range of [2^10, 2^11].

• @zehua2 @dreamsclogs
The number of stones is less than 1100 so pos will always be less than 2^11 (2048).
Stone positions could be theoretically up to 2^31 but k is practically not possible to be that big for the parameter as the steps must start from 0 and 1 and at the 1100th step the greatest valid k would be 1100. So combining pos and k is safe here.
(I have also added this comment to my post.)

• @mzchen Actually, the greatest valid k at the 1100th step is 1100.

• @mzchen I thought your pos is the position number of stone. I just noticed you are storing the index of stones, so you are right.

• @StefanPochmann
Ah, you are damn right! I got confused when I was writing the comment. Post updated.

• @mzchen Execuse me, I ran your "unordered_map memorization" solution and it took 62 ms. Did I do it wrong?

• @quesder it's not wrong. the non-memorizing version takes 9ms.

• What about [0,2]? Should we add `if (pos ==0 && k!=gap) return false;` after we got the gap?

• @dukeforever No. Why?