(1)鏃墮棿澶嶆潅搴?/p> (2)絀洪棿澶嶆潅搴?br>
絎簩璁?鎺掑簭綆楁硶
n杈冨ぇ 銆愬揩閫熸帓搴忋?br>n杈冨皬 銆愬啋娉℃帓搴忋?br>n杈冨ぇ n鍊艱緝灝?銆愯鏁版帓搴忋?br>鍙杗鐨勬渶鍊?銆愬爢鎺掑簭銆?br>n杈冨ぇ 瑕佹眰紼沖畾鎬?銆愬綊騫舵帓搴忋?br>
絎笁璁?綰挎ф暟鎹粨鏋?br>1.鏍?br>(1)DFS鐨勬樉寮忓啓娉?=> 綾諱技BFS
(2)鍥炴函 => DFS+鐘舵佽繕鍘?br>銆愭眰鎬繪柟妗堟暟鎴栬呮渶浼樻柟妗堥棶棰樸?br>2.闃熷垪
BFS
銆愭眰鏈灝戞搷浣滄鏁般?br>
絎洓璁?鏍戝艦緇撴瀯鐨勫簲鐢?br>1.浜屽弶鎺掑簭鏍?O(nlogn)
閫掑綊鏋勯?br>
2.鍝堝か鏇兼爲 => 鍫嗗疄鐜?br>
3.鏍戠姸鏁扮粍 => 閭繪帴琛?br>銆愯矊浼糔OIp瓚呯翰銆?
榪涗竴姝ョ殑緋葷粺鍖栧皢鍦ㄨ瀹岃繖鏈功鍚庯紝浠ヨ〃鏍兼垨鑰呮濈淮瀵煎浘鐨勫艦寮忓嚭鐜?
涓銆佷箻娉曞師鐞?/p>
涔樻硶鍘熺悊璁ㄨ鍒嗛樁孌靛姙浜嬭繃紼嬩腑鐨勮鏁伴棶棰?
鐢ㄩ泦鍚堣瑙傜偣瑙ｉ噴錛?br>鎶婂垎闃舵榪涜鐨勪簨鎯呯湅浣滀竴縐嶅閲嶉夊彇榪囩▼錛屾瘡涓涓繃紼嬮兘鏄嚜鏌愪釜闆嗗悎鎸戦変竴涓厓绱狅紝鐒跺悗鑰冭檻鏈夊灝戠涓嶅悓鐨勬寫閫夋柟寮?
銆愬簲鐢ㄣ戞暟鐨勬暣闄わ紝鍥犳暟鍒嗚В闂
銆愪緥3銆?160鐨勬綰︽暟涓暟.
2160 = 2^4 * 3^3 * 5^1
鍒欎換鎰忕害鏁板艦寮忎負2^i * 3^j * 5 ^k (0<=i<=4, 0<=j<=3, 0<=k<=1)
鎵浠ョ害鏉熶釜鏁頒負(4+1)(3+1)(1+1)=32.
銆愪緥5銆戜粠棰樼洰涓娊璞″嚭妯″瀷.
銆愪緥6銆?涓涓粨璁猴細[u,v]=n, 浠鐨勫洜瀛恟鐨勪釜鏁頒負n(r)錛屾湁k涓洜瀛?
鍒欑鍚堣姹傜殑鏁板涓暟涓?(2n(r)+1)*..*(2n(r)+1).
闇瑕佹敞鎰忕殑鏄渶澶у叕綰︽暟錛屾槸鍘諱袱鏁版煇鍥犲瓙鐨勬渶澶у?
銆愪緥7銆戣ˉ闆嗘濇兂.(鎺掗櫎娉?
鍦ㄦ暣闄ゆч棶棰樹腑錛岀‘瀹氬墠n-1浣嶏紝鐒跺悗鍒嗙被璁ㄨ絎琻浣嶇殑鎯呭喌.
銆愪緥8銆戠湅浜嗭紝鏈噦.
浜屻侀噸澶嶆帓鍒?/p>
銆愬畾涔夈戝鏋滃湪鍚屼竴涓猲闃墮泦鍚?鏈塶涓厓绱?涓緷嬈¤繘琛宬嬈￠夊彇錛岃屼笖閫夎繃鐨勫厓绱犺繕鍙互鍐嶉夛紝鍒欎竴鍏辨湁n^k涓笉鍚岀殑閫夊彇鏂瑰紡(鍗抽噸澶嶆帓鍒楁柟寮?.
銆愬簲鐢ㄣ戠┖闂磋В鏋愬嚑浣曘侀泦鍚堝瓙闆嗘ц川鐨勮璁?/p>
銆愪緥5銆戠珛鏂逛綋{(x,y,z):0<=x,y,z<=a}欏剁偣鍧愭爣.
璇?鏄劇劧姣忎釜欏剁偣(x,y,z)鐨勪笁涓潗鏍囬兘鏄泦鍚坽0,a}鐨勫厓绱狅紝鎵浠ュ叡鏈?^3涓笉鍚岀殑欏剁偣.鍧愭爣鐣?
銆愪緥6銆戝湪涓夌淮嬈ф皬絀洪棿緇欏嚭9涓牸鐐?鍧愭爣鍊間負鏁存暟)錛屾眰璇佸叾涓繀鏈変袱鐐逛腑鐐瑰潗鏍囦負鏍肩偣.
璇?鑻ヤ袱鐐笰(x1,y1,z1)鍜孊(x2,y2,z2)鐨勪腑鐐逛負鏍肩偣錛屽繀鏈墄1鍜寈2錛寉1鍜寉2錛寊1鍜寊2鍜屼負鍋舵暟錛屽嵆涓よ呭鍋舵х浉鍚?
鑰冭檻鑰冭檻嬈ф皬絀洪棿鏍肩偣濂囧伓鎬ф儏鍐靛彲鐭ワ紝姣忎釜鍧愭爣鍊奸兘鏄瘂濂囨暟錛屽伓鏁皚涓竴涓厓绱狅紝鎵浠ユ牸鐐規湁8縐嶄笉鍚岀殑濂囧伓鎬?浠庤岋紝鍘熷懡棰樼敱鎶藉眽鍘熺悊鍙瘉.
銆愪緥7銆憂闃墮泦鍚堝叡鏈?^n涓瓙闆嗭紝2^n-1涓湡瀛愰泦.
銆愪緥9銆戞鐩橀棶棰橈紝浠庡乏涓嬭璧板埌鍙充笂瑙掞紝n+1琛岋紝m+1鍒楋紝姹傝瘉f(m,n)<=2^(mn).
璇?閬撹礬灝嗗煄甯傚垎鎴恗n涓柟鍧楋紝鑰岃礬綰垮張灝嗘柟鍧楀垎鎴愪袱涓瓙闆?鍏朵腑涓涓彲鑳戒負絀?錛屾樉鐒朵笉鍚屽瓙闆嗕釜鏁頒負2^(mn).鍗砯(m,n)<=2^(mn)錛屽綋涓斾粎褰搈=n=1鏃剁瓑鍙鋒垚绔?
涓夈佹帓鍒?br>
銆愬畾涔夈戜粠n涓笉鍚岀殑鍏冪礌涓湁嬈″簭鍦伴夊彇k(1<=k<=n)涓紝鍙仛浠巒涓笉鍚屽厓绱犱腑鍙栧嚭k涓厓绱犵殑涓涓帓鍒?
銆愬簲鐢ㄣ戠粍鏁伴棶棰樹腑鐨?#8220;鏃犻噸澶嶆暟瀛?#8221;闂.
銆愪緥4銆戝埄鐢ㄨˉ闆嗘濇兂澶勭悊鏃犻噸澶嶆暟瀛楅棶棰?鑰冭檻涓嶅弬鍔犵粍鏁扮殑鏁板瓧鏁撮櫎鎯呭喌鍜屾墍鏈夋暟瀛楁暣闄ゆ儏鍐?鑱旂郴1.7)
銆愪緥5銆戦愪綅璁ㄨ.
銆愪緥7銆戞敞鎰忓埌k涓暟瀛楀彇鑷笉鍚岃鍒?鑱旀兂鐨囧悗闂)錛屾墍浠ュ瓙闆嗕釜鏁頒負k!錛屾瘡涓瓙闆嗙殑鍜屽垎琛屽垪璁ㄨ錛岀瘡鍔犲嵆鍙?
銆愪緥10銆戞湁2n涓漢鍙傚姞鏀跺彂鐢墊姤鍩硅錛屾瘡涓や漢緇撲負涓瀵逛簰鍙戜簰鏀訛紝鏈夊灝戠涓嶅悓鐨勭粨瀵規柟寮?(鎼厤闂)
瑙?(2n-1)(2n-3)...3*1=(2n)!/(2^n*n!)
闇瑕佹敞鎰忕殑鏄眰鍜屼嬌鐢ㄧ殑鎬濇兂: 鍏堟眰鍑哄叏閮ㄦ暟鐨勭Н錛岀劧鍚庡幓鎺夐噷闈㈢殑鍋舵暟.涔熸槸涓縐嶉棿鎺ョ殑鏂規硶.
鍥涖佸姞娉曞師鐞?br>
銆愰泦鍚堢殑鍒嗗垝銆?br>鑻ユ妸涓涓泦鍚圔鍒嗘垚涓浜涘瓙闆咮1,B2,...,Bk錛屼嬌寰?br>(i)B1∪B2∪...∪Bk = B;
(ii)B1∩B2=∅ ,...錛孊k-1∩Bk=∅.
婊¤凍榪欎袱鏉℃ц川鐨勫瓙闆咮1,B2,...,Bk錛屽彨鍋欱鐨勪竴涓垎鍒?
銆愬畾涔夈?span style="BACKGROUND-COLOR: #c0c0c0"> |B|=|B1|+|B2|+...+|Bk| 鍔犳硶鍏紡
榪欑閫氳繃鍒嗗垝璁℃暟鐨勫師鐞嗗彨鍋氬姞娉曞師鐞?
銆愪緥1銆戠幇鏈夐暱搴﹀垎鍒負1,2,...,n鐨勭粏鏈ㄦ鍚勪竴鏍癸紝鍙互浠ュ畠浠負杈規瀯鎴愬灝戠涓嶅悓鐨勪笁瑙掑艦?
瑙?浠鐨勯暱搴﹀榪欎簺涓夎褰㈠垎綾伙紝灝哻=k鐨勪笁瑙掑艦闆嗗悎璁板仛Bk錛屽垯鏋勬垚浜嗛泦鍚圔鐨勪竴涓垎鍒?
鍦˙k涓紝涓夎褰笁杈瑰垎鍒負a<b<k錛屽叾涓璳涓哄畾鍊鹼紝浜庢槸鍙皢(a,b)瀵瑰簲浜庡鉤闈腑鐨勬牸鐐?閫氳繃闄愬埗鏉′歡a<k,b<k,a+b>k鎴戜滑鍙互鐭ラ亾錛岀鍚堟潯浠剁殑鏍肩偣鍦╝=b,a+b=k,b=k涓夋潯鐩寸嚎鍥存垚鐨勪笁瑙掑艦鍐?/span>,
鎵浠ヨ嫢k涓哄伓鏁幫紝鏈墊Bk|=1/4*(k-2)^2
鑻涓哄鏁幫紝鏈墊Bk|=1/4(k-1)(k-3)
浠庤屽彲浠ユ眰寰梶B|.(浜岄樁絳夊樊鏁板垪姹傚拰涓嶇啛)
銆愪緥2銆戞眰鏂圭▼x^2-[x]^2=(x-x[x])^2鍦ㄥ尯闂碵1,n]涓牴鐨勬暟鐩?
瑙?灝嗗尯闂碵k,k+1)涓牴鐨勯泦鍚堣鍋欱k.
鑻∈[k,k+1)錛岃k=[x],p=x-[x](0<=p<1)錛屽彲寰?kp=[2kp+p^2].
鏄劇劧絳夊紡涓よ竟涓烘暣鏁幫紝鎵浠=0,1/2k,...,2k-1/2k錛屾晠鑰寍Bk|=2k銆?br>鐢卞姞娉曞師鐞嗗彲鐭ワ紝|B|=n^2-n+1
浜斻佸甫闄愬埗鏉′歡鐨勬帓鍒楅棶棰?br>
(i) 闂存帴鏂規硶
銆愭帓闄ゆ硶銆戝厛鍋囧畾涓嶅瓨鍦ㄩ檺鍒舵潯浠訛紝姹傚嚭鎵鏈夋儏鍐電殑鏁扮洰錛涘啀鑰冭檻鍙楀埌闄愬埗鏉′歡錛岃屼笉鍏佽鍑虹幇鐨勬儏鍐墊暟鐩?
(ii) 鐩存帴鏂規硶
銆愪紭闄愭硶銆戜紭鍏堣В鍐沖彈闄愬璞?鍙楅檺瀵硅薄鎴栧彈闄愬厓绱?鐨勫畨緗紝鐒跺悗鍐嶈冭檻涓鑸璞＄殑瀹夌疆闂.
銆愭彃鍏ユ硶銆戦鍏堝畨鎺掍竴鑸厓绱狅紝鐒跺悗灝嗛鍏堝厓绱犳彃鍏ュ埌鍏佽鐨勪綅緗笂.(鏌愪簺鍏冪礌鐩擱偦鎴栬呬笉鐩擱偦)
銆愯涓娉曘戦鍏堣鎶婅姹傜浉閭繪帓鍒楃殑鍏冪礌鐪嬫垚涓涓暣浣擄紝鍚屽叾浠栧厓绱犱竴鍚屾帓鍒楋紝鐒跺悗鍐嶈冭檻榪欎釜鏁翠綋鍐呴儴鐨勫厓绱犻棿鐨勬帓鍒楅棶棰?
銆愪緥8銆?0涓妭鐩腑鏈?涓紨鍞憋紝4涓垶韞堬紝瑕佹眰姣忎袱涓垶韞堜箣闂磋嚦灝戝畨鎺掍竴涓紨鍞?
瑙?鍙嶈繃鏉ョ湅闂錛屽師鍛介絳変環浜庡湪浠繪剰涓ゆ紨鍞?杈圭晫鎯呭喌鐨勮瘽涓涓?涓畨鎺掓垨涓嶅畨鎺掍竴涓垶韞堬紝鑰岃繖鏍風殑鍙兘浣嶇疆鍏?涓?鎵浠ュ叡6!*P(7,4)縐嶉『搴?
鍏佺粍鍚?br>
銆愬畾涔夈戜粠n涓笉鍚岀墿浠朵腑鏃犳搴忓湴(涓嶈欏哄簭鍦?閫夊彇k涓紝鍙仛浠巒涓墿浠朵腑閫塳涓殑涓涓粍鍚?
濡傛灉鑰冭檻k涓墿浠剁殑閫夊彇欏哄簭錛屽彲寰桺(n,k)=C(n,k)*k!
浠庤屽緱鍒扮粍鍚堢殑璁＄畻鍏紡C(n,k)=n!/k!(n-k)!
(i)C(n,n-k)=C(n,k)
(ii)C(n,k)+C(n,k-1)=C(n+1,k)